local-field eﬀects and nanostructuring for controlling ...€¦ · terials, local-ﬁeld-induced...

Local-Field Effects and Nanostructuring

for Controlling Optical Properties

and Enabling Novel Optical Phenomena

by

Ksenia Dolgaleva

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Robert W. Boyd

The Institute of OpticsThe College

School of Engineering and Applied Sciences

University of RochesterRochester, New York

2008

ii

Curriculum Vitae

Ksenia Dolgaleva was born in Astrakhan, Russia, on November 2, 1977. She grad-

uated from a high school with a silver medal, and entered an undergraduate program

at the Department of Physics of Moscow State University in 1994. She joined the

research group of Prof. Oleg Nanii in 1998. Under supervision of Prof. Nanii, she

theoretically investigated continuous-wave two-color generation in a laser with a dis-

persive cavity. While in Prof. Nanii’s research group, Ksenia gave three presentations

at two international conferences on Optics held in St. Petersburg, Russia, and wrote

an article, published in Quantum Electronics journal. She graduated from Moscow

State University with a M. S. degree in Physics in January 2000. She was awarded a

prize for an outstanding Master thesis from the Russian Physical Society.

Ksenia entered the Ph. D. program at Moscow State University in April 2000. At

the same time, she worked for the Optictelecom company, which was collaborating

with the research group of her advisor, Prof. Nanii. Her job at Optictelecom involved

preparation of scientific seminars and a lecture course, and co-authoring a manual

on Fiber Optics for Optictelecom employees and undergraduate students of Moscow

State University.

Ksenia moved to Rochester, NY and entered the Ph. D. program at the Insti-

tute of Optics, University of Rochester, in September 2001. She started her research

under the supervision of Dr. Andrey Okishev at the Laboratory for Laser Energet-

ics. Her first project was related to the characterization of laser performance of

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a new crystal, Yb:GdCOB, a potential candidate for broadband amplifiers for the

60-beam OMEGA laser facility. After two years at the Laboratory for Laser Energet-

ics, she joined Prof. Robert Boyd’s research group in June 2003. Under supervision

of Prof. Boyd, Ksenia worked on various projects, including composite laser ma-

terials, local-field-induced microscopic cascading in nonlinear optics, optical activity

from artificial chiral structures, cholesteric liquid crystal lasers, and single-step phase-

matched third-harmonic generation in a 1D photonic crystal. While in Prof. Boyd’s

group, Ksenia co-authored 11 papers, published and in preparation for publication

in refereed journals, and one encyclopedia article. She gave presentations at 8 in-

ternational conferences in Optics, and co-authored 4 more conference presentations

given by other researchers. Ksenia was awarded a prize for outstanding student pre-

sentation at the Frontiers in Optics OSA Annual Meeting in Rochester in October

2008.

iv

Acknowledgements

I am endlessly grateful to my advisor, Prof. Robert Boyd, for his great help,

guidance, and wise supervision. He contributed a lot towards my growth as a young

scientist, and I cannot think of a better experience than that I gained being a part of

his research group.

I would like to acknowledge Prof. Peter Milonni for his mentoring, support, and

belief in success of some of my projects that I had been struggling through and that

eventually worked.

I am also thankful to Prof. John Sipe for his mentoring, valuable discussions, and

healthy criticism that ensured success and clarity of some crucial parts of my work.

I am grateful to Dr. Svetlana Lukishova for her support, help and encouragement of

my efforts and struggling. Her good advice helped me to make some crucial decisions

that cardinally changed my future towards the best.

I would like to thank Prof. Shaw Horng Chen and his student Ku-Hsien (Simon)

Wei for a great opportunity to collaborate with them on a wonderful project, and to

realize how useful my skills and knowledge could be for someone working in a different

field.

I am also grateful to Dr. Sergei Volkov, Prof. Martti Kauranen and Prof. Yuri

Svirko for a great collaboration and valuable discussions in the process of which I

have learned a lot about a field that is new to me.

I would like to acknowledge Prof. Paras Prasad, Prof. Ion Tiginyanu, Dr. Paul

v

Markowicz, and Dr. Lilian Syrbu for fruitful collaborations.

I would like to thank Prof. Carlos Stroud, Prof. Mikhail Noginov, Prof. John Dowl-

ing, Prof. Lukas Novotny, Prof. Nick Lepeshkin, Prof. Stephen Rand, and Prof. An-

thony Siegman for valuable discussions.

I am thankful to Prof. Lewis Rothberg and Prof. Miguel Alonso for agreeing to

serve on my Ph. D. committee and for their advice on how to improve my Ph. D.

thesis.

Many thanks to Dr. Andrey Okishev and Dr. John Zuegel for valuable laboratory

skills that I have gained while working with them.

I am grateful to Dr. Frank Duarte, Dr. Robert James, and Prof. Steve Jacobs

for assisting in my attempt to prepare a new optical material that is interesting to

investigate from the perspective of my research.

I would like to thank fellow graduate students and postdocs from Prof. Boyd’s

research group. I am especially grateful to Heedeuk Shin and Andreas Liapis who

collaborated with me directly. Many thanks to Giovanni Piredda, Aaron Schweins-

berg, Colin O’Sullivan-Hale, Petros Zerom, George Gehring, Anand Jha, Zhimin Shi,

Joseph Wornehm, Mehul Malik, Cliff Chen, Hye Jeong Chang, and Sandrine Hocde.

I am grateful to Per Adamson and Brian McIntyre for their technical assistance

and help with equipment. I am thankful to the Institute of Optics administrative as-

sistants: Maria Schnitzler, Joan Christian, Lissa Cotter, Gina Kern, Noelene Votens,

Betsy Benedict, and Maria Banach. Without their help I would not succeed to com-

vi

plete my program.

I would especially like to acknowledge the Director of the Institute of Optics,

Prof. Wayne Knox, for his support during my first years at the Institute and for

being such a dedicated Director whose contribution towards the development of the

Institute is hard to overestimate.

I am endlessly grateful to my supportive family: my husband Sergei, my mother

Natalia Feketa, my daughter Natalia, my step-father Iosif Feketa, my father Pavel

Dolgalev, my sister Anna Feketa, my aunts Irina Dolgaleva, Valentina Fomina, and

Lubov Anpilova, my cousins and my grandmother Ekaterina Sorokina. I would like

to devote this thesis to them all, as without their support I would not be able to get

through this challenge.

Many thanks to my supportive friends.

vii

Publications

1. K. Dolgaleva, H. Shin. R. W. Boyd, and J. E. Sipe, “Observation of microscopiccascaded contribution to the fifth-order nonlinear susceptibility,” to be submitted toPhys. Rev. Lett.

2. K. Dolgaleva, R. W. Boyd, and P. W. Milonni, “Effects of Local Fields on LaserGain for Layered and Maxwell Garnett Composite Materials,” accepted forpublicaiton in J. Phys. A.

3. S. N. Volkov, K. Dolgaleva, R. W. Boyd, K. Jefimovs, J. Turunen, Y. Svirko, B. K.Canfield, and M. Kauranen, “Optical activity in diffraction from a planar array ofachiral nanoparticles,” submitted to Phys. Rev. Lett.

4. S. K. H. Wei, S. H. Chen, K. Dolgaleva, S. G. Lukishova, and R. W. Boyd, “Robustorganic lasers comprising glassy-cholesteric pentafluorene doped with a red-emittingoligofluorene,” submitted to Appl. Phys. Lett.

5. K. Dolgaleva, S. K. H. Wei, S. G. Lukishova, S. H. Chen, K. Schwertz, and R. W.Boyd, “Enhanced laser performance of cholesteric liquid crystals doped witholigofluorene dye,” J. Opt. Soc. Am. B 25, 1496–1504 (2008).

6. K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused bylocal-field effects in the two-level atom,” Phys. Rev. A 76, 063806 (2007).

7. K. Dolgaleva and R. W. Boyd, “Laser gain media based on nanocompositematerials,” J. Opt. Soc. Am. B 24, A19–A25 (2007).

8. K. Dolgaleva, R. W. Boyd, and P. W. Milonni, “Influence of local-field effects on theradiative lifetime of liquid suspensions of Nd:YAG nanoparticles,” J. Opt. Soc. Am.B 24, 516–521 (2007).

9. L. Syrbu, V. V. Ursaki, I. M. Tiginyanu, K. Dolgaleva, and R. W. Boyd, “Red andgreen nanocomposite phosphors prepared from porous GaAs templates,” J. Opt. A:Pure Appl. Opt. 9, 401–404 (2007).

10. L. Sirbu, V. V. Ursaki, I. M. Tiginyanu, K. Dolgaleva, and R. W. Boyd, “Er- andEu-doped CaP-oxide porous composites for optoelectronic applications,” Phys. Stat.Sol. (RRL) 1, R13–R15 (2007).

11. P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, P. N. Prasad, K.Dolgaleva, N. N. Lepeshkin, and R. W. Boyd, “Enhancement of third-harmonicgeneration in a polymer-dispersed liquid-crystal grating,” Appl. Phys. Lett. 87,051102 (2005).

12. V. G. Voronin, K. P. Dolgaleva, and O. E. Nanii, “Two-color generation in asolid-state laser with a dispersive cavity,” Quantum Electron. 30, 778–782 (2000).

viii

Other Publications

1. A. V. Okishev, K. P. Dolgaleva, and J. D. Zuegel, “Experimental optimization ofdiode-pumped Yb:GdCOB laser performance for broadband amplification at 1053nm,” ICONO/LAT conference proceedings (St.-Petersburg, Russia, 2005).

2. K. Dolgaleva, N. Lepeshkin, and R. W. Boyd, “Frequency doubling,” Encyclopediaof Nonlinear Science, (London, Routlidge, 2004).

3. S. K. Isaev, V. G. Voronin, O. E. Nanii, A. N. Turkin, K. P. Dolgaleva, D. D.Scherbatkin, “Fundamentals of Fiber Optics and Integral Optics” (anundergraduate-level manual, in Russian) Moscow State University, 2002.

Conference Papers

1. K. Dolgaleva, H. Shin, R. W. Boyd, and J. E. Sipe, “Experimental separation ofmicroscopic cascading induced by local-field effects,” Frontiers in Optics and LaserScience (FiO/LS) 2008, Rochester, NY (USA).

2. R. W. Boyd, K. Dolgaleva, G. Piredda, and A. Schweinsberg, “Nonlinear optics ofstructured photonic materials,” Frontiers in Optics and Laser Science (FiO/LS)2008, Rochester, NY (USA).

3. K. Dolgaleva, R. W. Boyd, S. N. Volkov, K. Jefimovs, J. Turunen, Y. Svirko, B. K.Canfield, and M. Kauranen, “Polarization changes in diffraction from planarperiodic patterns with pure structural and molecular chirality,” QuantumElectronics and Laser Science (QELS) 2008, San Jose, CA (USA).

4. K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Microscopic cascading in fifth-ordernonlinearity induced by local-field effects,” Nonlinear Photonics 2007, Quebec, QC(Canada).

5. K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascade-like nonlinearity caused bylocal-field effects: Extending Bloembergen’s result,” Quantum Electronics and LaserScience (QELS) 2007, Baltimore, MD (USA).

6. K. Dolgaleva, K. H. S. Wei, A. Trajkovska, S. Lukishova, R. W. Boyd, and S. H.Chen, “Oligofluorene as a new high-performance dye for cholesteric liquid crystallasers,” Frontiers in Optics and Laser Science (FiO/LS) 2006, Rochester, NY (USA).

7. S. K. H. Wei, K. Dolgaleva, A. Trajkovska, S. Lukishova, R. W. Boyd, and S. H.Chen, “High performance and color purity in oligofluorene-doped cholesteric liquidcrystal laser,” Frontiers in Optics and Laser Science (FiO/LS) 2006, Rochester, NY(USA).

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8. K. Dolgaleva and R. W. Boyd, “A bi-exponential fluorescence decay in compositematerials based on Nd:YAG nanoparticles,” Conference on Lasers andElectro-Optics (CLEO) 2006, Long Beach, CA (USA).

9. K. Dolgaleva and R. W. Boyd, “Influence of local-field effects on the radiativeproperties of Nd:YAG nanoparticles in a liquid suspension,” American PhysicalSociety (APS) March Meeting 2006, Baltimore, MD (USA).

10. A. V. Okishev, K. P. Dolgaleva, and J. D. Zuegel, “Experimental optimization ofdiode-pumped Yb:GdCOB laser performance for broadband amplification at 1053nm,” International Conference on Coherent and Nonlinear Optics collocated withInternational Conference on Lasers, Applications, and Technologies (ICONO/LAT),St.-Petersburg, Russia, 2005.

11. P. P. Markowicz, V. Hsiao, H. Tiryaki, P. N. Prasad, N. N. Lepeshkin, K. Dolgaleva,and R. W. Boyd, “Enhancement of third-harmonic generation in Photonicscrystals,” Conference on Lasers and Electro-Optics (CLEO) 2005, Baltimore, MD(USA).

12. K. Dolgaleva, N. Lepeshkin, R. W. Boyd, P. P. Markowicz, V. K. S. Hsiao,H. Tiryaki, A. N. Cartwright, and P. N. Prasad, “Phase-matched third-harmonicgeneration in a 1-D Photonics crystal,” Frontiers in Optics and Laser Science(FiO/LS) 2004, Rochester, NY (USA).

13. O. E. Nanii, K. P. Dolgaleva, “A novel construction of a two-color laser with adispersive cavity using two pump sources,” International Optical Congress“Optics–XXI century,” St.-Petersburg, Russia, 2000.

14. K. P. Dolgaleva, K. N. Belov, O. E. Nanii, “Two-color generation in a laser with adispersive cavity and two pump sources,” International Conference of YoungScientists and Specialists “Optics–2000,” St.-Petersburg, Russia.

15. K. P. Dolgaleva, “Wavelength competition in a two-color laser with a dispersivecavity,” International Conference of Young Scientists and Specialists “Optics–99,”St.-Petersburg, Russia, 1999.

x

Abstract

My Ph. D. thesis is devoted to the investigation of the methods for controlling

and improving the linear and nonlinear optical properties of materials. Within my

studies, two approaches are considered: nanostructuring and invoking local-field ef-

fects. These broad topics involve various projects that I have undertaken during my

Ph. D. research.

The first project is on composite laser gain media. It involves both nanostructuring

and using local-field effects to control the basic laser parameters, such as the radiative

lifetime, small-signal gain and absorption, and the saturation intensity. While being

involved in this project, I have performed both theoretical and experimental studies

of laser characteristics of composite materials. In particular, I have developed simple

theoretical models for calculating the effective linear susceptibilities of layered and

Maxwell Garnett composite materials with a gain resonance in one of their compo-

nents. The analysis of the results given by the models suggests that local-field effects

provide considerable freedom in controlling the optical properties of composite laser

gain media. I have also experimentally measured the radiative lifetime of Nd:YAG

nanopowder suspended in different liquids to extract information regarding local-field

effects.

The second project is devoted to the investigation of a not-well-known phenomenon

that local-field effects can induce, which is microscopic cascading in nonlinear optics.

This project involves the theoretical prediction of local-field-induced microscopic cas-

xi

cading effect in the fifth-order nonlinear response and its first experimental obser-

vation. This effect has been mostly overlooked or underestimated, but could prove

useful in quantum optics. I have shown that, under certain conditions, the microscopic

cascaded contribution can be a dominant effect in high-order nonlinearities.

The third project is about characterization of laser performance of a new dye,

oligofluorene, embedded into cholesteric liquid crystal (CLC) structures. These struc-

tures constitute self-assembling mirrorless distributed-feedback low-threshold laser

systems. I have performed a detailed comparative experimental study of the laser

characteristics of cholesteric liquid crystals doped with oligofluorene and a well stud-

ied dye, DCM, commonly used for lasing in CLCs. I have experimentally demon-

strated that oligofluorene-doped CLCs yield a total output energy in the transverse

single-mode regime five times that of DCM-doped CLCs with superior temporal and

spatial stability.

In the fourth project I investigated the polarization changes of light diffracted

from artificial planar chiral structures. I have performed the measurements of the po-

larization state of light diffracted from planar arrays of nanoparticles with molecular

and pure structural chirality. Both sorts of samples are shown to lead to comparable

polarization changes in the diffracted light. The results can be explained by a simple

model in which the polarization effects resulted from an independent scattering by

individual particles, with no interparticle coupling, where the array structure simply

determines the direction of the diffraction maximum. It invites a conclusion that

xii

structural and molecular chirality are indistinguishable in diffraction experiments, in

contrast to some earlier published statements.

Contents

1 Background 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Composite Optical Materials . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Composite Geometries . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Local Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Local-Field Models for Homogeneous Media . . . . . . . . . . 8

1.3.2 Local Field and Cascading in Nonlinear Optics . . . . . . . . . 14

1.4 Lorentz–Maxwell–Bloch Equations . . . . . . . . . . . . . . . . . . . 17

1.5 Dye-Doped Cholesteric Liquid Crystal Lasers . . . . . . . . . . . . . . 20

1.6 Planar Chiral Structures . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Composite Laser Materials 27

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 The Idea of Composite Lasers . . . . . . . . . . . . . . . . . . . . . . 28

xiii

CONTENTS xiv

2.2.1 Influence of Local-Field Effects on Laser Properties of Dielectric

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Influence of Local-Field Effects on the Radiative Lifetime of Liquid

Suspensions of Nd:YAG Nanoparticles . . . . . . . . . . . . . . . . . 40

2.3.1 Sample Preparation and Experiment . . . . . . . . . . . . . . 44

2.3.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4 Optimization of Laser Gain Properties for Layered and Maxwell Gar-

nett Composite Geometries . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.1 Lorentz local field in a resonant medium . . . . . . . . . . . . 55

2.4.2 Linear Susceptibility of Layered Composite Materials . . . . . 60

2.4.3 Linear Susceptibility of Maxwell Garnett Composite Materials 65

2.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Microscopic Cascading 84

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2 Theoretical Prediction of Microscopic Cascading . . . . . . . . . . . . 87

3.2.1 Maxwell–Bloch equations approach . . . . . . . . . . . . . . . 87

3.2.2 The Naıve Local-Field Correction . . . . . . . . . . . . . . . . 90

3.2.3 Bloembergen’s approach . . . . . . . . . . . . . . . . . . . . . 93

3.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

CONTENTS xv

3.4 Experimental Observation of Microscopic Cascading . . . . . . . . . . 106

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4 Dye-Doped CLC Lasers 122

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.4 Stability and Frequency Mode Competition . . . . . . . . . . . . . . . 131

4.5 Lasing Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.5.1 Transverse single-mode regime . . . . . . . . . . . . . . . . . . 134

4.5.2 Transverse multi-mode regime . . . . . . . . . . . . . . . . . . 137

4.5.3 Laser Output Degradation Issues in CLCs . . . . . . . . . . . 140

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5 Artificial Chiral Structures 144

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.3 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.4 Wire Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6 Summary 158

Bibliography 163

CONTENTS xvi

A Maxwell Garnett Mesoscopic Electric Field 178

B Optical Bistability 181

List of Figures

1.1 Composite material geometries . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Basic laser parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 SEM picture of Nd:YAG nanopowder . . . . . . . . . . . . . . . . . . 44

2.3 Liquids used for suspending Nd:YAG nanopowder . . . . . . . . . . . 45

2.4 Experimental setup for measuring the radiative lifetime of Nd:YAG

nanopowder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5 Typical fluorescence decay in the Nd:YAG nanopowder . . . . . . . . 47

2.6 Experimentally measured radiative lifetimes of the Nd:YAG-nanopowder

suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.7 Small-signal gain of a layered composite material as a function of the

refractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


volume fraction of the resonant component . . . . . . . . . . . . . . . 74

xvii

LIST OF FIGURES xviii


detuning of the optical field . . . . . . . . . . . . . . . . . . . . . . . 76

2.10 Small-signal gain of a Maxwell Garnett composite material as a func-

tion of the refractive index of the non-resonant host . . . . . . . . . . 78


tion of the inclusion volume fraction . . . . . . . . . . . . . . . . . . . 79


tion of the detuning of the optical field . . . . . . . . . . . . . . . . . 80

3.1 The ranges of validity of various susceptibility models . . . . . . . . . 101

3.2 Real and imaginary parts of the total susceptibility . . . . . . . . . . 103

3.3 The ratio of the absolute values of cascaded and direct contributions

to χ(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.4 Real and imaginary parts of the direct and cascaded contributions to

χ(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.5 Experimental setup for degenerate four-wave mixing . . . . . . . . . . 108

3.6 Intensities of first and second diffracted orders on the logarithmic scale 110

3.7 Typical experimental data for χ(3) and χ(5) . . . . . . . . . . . . . . . 111

3.8 Phase matching diagram . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.9 Normalized efficiencies of contributions to χ(5) . . . . . . . . . . . . . 116

3.10 Measured fifth-order nonlinear susceptibility and its macroscopic cas-

caded contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

LIST OF FIGURES xix

4.1 Reflectance and lasing spectra of dye-doped CLC structures . . . . . 128

4.2 Experimental setup for lasing in CLC . . . . . . . . . . . . . . . . . . 129

4.3 Intensity distribution of the laser output of dye-doped CLC lasers . . 131

4.4 A photograph of the lasing output of dye-doped CLC structures . . . 134

4.5 Laser output from dye-doped CLC structures in single-mode regime . 135

4.6 Slope efficiency of dye-doped CLC lasers in single-mode regime . . . . 136

4.7 Laser output from dye-doped CLC structures in multi-mode regime . 137

4.8 Slope efficiency of dye-doped CLC lasers in multi-mode regime . . . . 138

5.1 Experimental setup and sample layout . . . . . . . . . . . . . . . . . 148

5.2 Polarization azimuth rotation and ellipticity of light in diffraction from

planar structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.3 Scattering from a tilted cross . . . . . . . . . . . . . . . . . . . . . . 153

5.4 Polarization azimuth rotation predicted by wire model . . . . . . . . 154

B.1 Parameter space for bistability in population inversion . . . . . . . . 184

Chapter 1

Background

1.1 Introduction

Optical materials have a broad range of applications in a variety of aspects of human

life. Among those are medicine, military, communications, computing, manufactur-

ing, and various industrial applications. Rapid progress of nanotechnology opens new

opportunities in designing optical materials with improved optical properties.

The primary goal of my Ph.D. dissertation is to investigate novel methods of con-

trolling linear and nonlinear optical properties of materials, and to come up with

new efficient solutions of how one can achieve the desired control. There are many

approaches one can use to significantly modify the optical properties of materials.

Among the methods I have been studying are nanostructuring and exploiting local-

field effects. The former method is a popular way to approach the problem of control-

1

CHAPTER 1. BACKGROUND 2

ling the optical properties of materials. With rapid development of nanotechnology,

this method becomes more and more important and accessible. The latter method,

relying on local-field effects, requires a deep understanding of fundamental nature of

the local-field phenomenon in both linear and nonlinear optics. If one gains the nec-

essary understanding, local-field effects will open many unknown opportunities and

serve as a source for novel unusual effects.

The two approaches to controlling the optical properties of nonlinear and laser

materials led to several projects which I have undertaken as a part of my Ph. D.

research. Among these projects are

• Optimization of laser properties of composite laser gain media through use of

local-field effects;

• Microscopic cascaded phenomena in nonlinear optics induced by local-field ef-

fects;

• Characterization and optimization of laser performance of dye-doped cholesteric

liquid crystal lasers;

• Polarization changes in diffraction off artificial planar chiral structures.

In the current chapter, I give an overview of various concepts that form the ba-

sis of my research. I describe the individual research projects in the latter chapters

of my dissertation. In Chapter 2, I present my theoretical and experimental work

on controlling the basic laser parameters of nanocomposite optical materials. This


study involves local-field effects in combination with nanostructuring. Chapter 3 is

devoted to a microscopic cascading phenomenon in nonlinear optics that I acciden-

tally uncovered theoretically, and later observed experimentally. This phenomenon is

a consequence of local-field effects. It has potential in applications for quantum and

conventional lithography. I present my work on characterization and optimization of

laser performance of dye-doped cholesteric liquid crystal lasers in Chapter 4. Chap-

ter 5 describes my experiment on polarization control of light diffracted from artificial

planar chiral structures. Finally, I summarize my Ph.D. research in Chapter 6

1.2 Composite Optical Materials

Nanocomposite optical materials are nanoscale mixtures of two or more homogeneous

constituents in which the individual particles are much smaller than the optical wave-

length, but still large enough to have their own dielectric identities. The optical

properties of composite materials can be adjusted by controlling the constituents and

morphology of the composite structure. Properly tailored composites can display the

best qualities of each of their constituents, or, in certain cases, can display proper-

ties that even exceed those of their constituents. These features render composite

materials valuable for applications in photonics and laser engineering.

Nanocomposite optical materials are becoming more and more important in laser

applications as nanofabrication technology has been rapidly developing. In particular,

nanoscale ceramic composite laser gain media with improved optical properties have


been reported [1, 2]. It has also been shown that one can improve the performance

of a laser material by mixing it with some other material on a nanoscale in such a

way that the thermal refractive index changes of the resulting composite material are

smaller than those of either of the constituents [3, 4]. In the current work we are

concerned with a somewhat different approach to controlling the laser properties of

nanocomposite materials: by implementing local-field effects [5, 6].

The optical properties of composite materials have been the subject of many

studies (see, for example, [7–10]). In particular, the modification of the radiative

lifetime of composite materials caused by local-field effects was addressed in many

publications both theoretically [11–15] and experimentally [16–24]. The influence

of the local-field effects on the nonlinear optical properties of composite materials

is even more significant, as the material response scales as several powers of the

local-field correction factor (i. e., the quantity equal to the ratio of the local field

acting on a typical emitter to the average field in the medium). Theoretical modeling

of the nonlinear optical response has been reported for many different geometries of

composite materials [25–28]. In particular, rigorous theories for Maxwell Garnett-type

composite materials [25] and layered composite materials [27] have been developed. It

was shown that a significant enhancement of the nonlinear optical response is possible

under certain conditions. A number of experiments in the field yielded promising

results [29–32]. Thus, a composite-material approach has proven to be a valuable

tool in designing optical materials with enhanced nonlinear response.


While there have been numerous studies of the nonlinear optical properties of

composite materials, there has not been a systematic study yet of their laser prop-

erties. In Chapter 2 we present a study of the influence of the local-field effects on

the laser properties of composite materials. We believe that the analysis done in this

work will help in further development of new materials for laser applications.

1.2.1 Composite Geometries

There are three types of composite geometries mainly discussed in the literature:

Maxwell Garnett composites [7, 8, 28], Bruggeman composites [32–34], and layered

composites [27,28,31] (see Fig. 1.1).

�h

�i

�a�b

�1

�2

�a� �b� �c�Figure 1.1: Composite material structures: (a) Maxwell Garnett geometry; (b)

Bruggeman geometry; (c) layered geometry.

The Maxwell Garnett type of composite geometry is a collection of small particles

(the inclusions) distributed in a host medium. The inclusions are assumed to be

spheres or ellipsoids of a size much smaller than the optical wavelength; the distance

between them must be much larger than their characteristic size and much smaller

than the optical wavelength. Under these conditions, one can treat the composite


material as an effective medium, characterized by an effective (average) dielectric

constant, ǫeff , which satisfies the relation [7, 8]

ǫeff − ǫh

ǫeff + 2ǫh

= fiǫi − ǫh

ǫi + 2ǫh

. (1.1)

Here ǫh and ǫi are the dielectric constants of the host and inclusion materials, respec-

tively, and fi is the volume fraction of the inclusion material in the composite.

In the Maxwell Garnett model the composite medium is treated asymmetrically.

It is assumed that the host material completely surrounds the inclusion particles, and

the result for the effective dielectric constant of the composite will be different if we

interchange the host and inclusion dielectric constants in the expression (1.1). This

problem is eliminated in the Bruggeman composite model [35], in which each particle

of each constituent component is considered to be embedded in an effective medium

characterized by ǫeff . The corresponding equation defining the effective dielectric

constant thus has the form [32]

0 = fa

ǫa − ǫeff

ǫa + 2ǫeff

+ fb

ǫb − ǫeff

ǫb + 2ǫeff

. (1.2)

Here ǫa and ǫb are the dielectric constants of the constituent components a and b and

fa and fb are the volume fractions of the components.

The third composite model shown in Fig. 1.1 is a layered structure consisting

of alternating layers of two materials (a and b) with different optical properties.


The thicknesses of the layers should be much smaller than the optical wavelength.

Materials of this sort are anisotropic. For light polarized parallel to the layers of such

a composite material the effective dielectric constant is given by a simple volume

average of the dielectric constants of the constituents:

ǫeff = fa ǫa + fb ǫb. (1.3)

The electric field in this case is spatially uniform, as the boundary conditions require

continuity of its tangential part on the border between two constituents. However,

for the light polarized perpendicular to the layers, the effective dielectric constant is

given by

1

ǫeff

=fa

ǫa

+fb

ǫb

. (1.4)

In the latter case the electric field is non-uniformly distributed between the two

constituents in the composite, and local-field effects are of particular interest.

The composite geometries that we described above are those used most often in

the design of composite optical materials.

1.3 Local Field

It is well-known that the field driving an atomic transition in a material medium, the

local field, is different in general from both the external field and the average field

inside the medium. The difference from the average field does not play a significant


role when one considers a low-density medium. To describe the optical properties

of such a system, one can use the macroscopic (ensemble average) field. However, if

the atomic density of a system exceeds ≈ 1015 cm−3 [36], the influence of local-field

effects becomes significant and cannot be neglected.

In order to account for local-field effects on the optical properties of a material, one

needs to apply a proper model relating the local field to its macroscopic counterparts,

namely the average field and polarization. The choice of the model strongly depends

on the medium of interest. For example, local field in a homogeneous medium can be

related to the macroscopic average field according to

Eloc = LE, (1.5)

where L is the local-field correction factor. The tilde denotes quantities oscillating

at an optical frequency. The relationship between the local and average fields can be

rather complicated in composite materials. In this Section we describe several models

of local field for homogeneous materials. Local field of composite geometries is a more

specific subject, and we cover it in Chapter 2.

1.3.1 Local-Field Models for Homogeneous Media

Existing theoretical models predict different expressions for the local-field correction

factor L, relating the local field to the average field in the medium (see Eq. (1.5)).

Two models, most commonly used to describe the experimental outcomes, are the


virtual-cavity model, or Lorentz model [37], and the real-cavity model [11]. A more

precise version of the real-cavity model is the Onsager model of local field [38]. Below

we give a brief description to these models of local field, and mention some other, more

sophisticated models for homogeneous media, which reconcile the simpler models.

Lorentz Local Field

It is conventional to describe local-field effects in a homogeneous material medium

using the well-known Lorentz model. In the simplest version of this model, one treats

the medium as a cubic lattice of point dipoles of the same sort. In order to find

the local field acting on a typical dipole of the medium, one surrounds the dipole of

interest with an imaginary spherical cavity of radius much larger than the distance

between the dipoles, and much smaller than the optical wavelength. The contributions

to the local field from the dipoles situated within the spherical cavity are accounted

for exactly, while the dipoles outside the cavity are treated as uniformly distributed,

characterized by some average macroscopic polarization. This approach yields the

well-known expression

Eloc = E +4π

3P (1.6)

for the local field Eloc in terms of the average macroscopic field E and the macroscopic

polarization P. Local field given by Eq. (1.6) is called “Lorentz local field.” It is

derived in many textbooks (see, for example, [5, 6]). The textbook model used for

deriving Eq. (1.6) is known as virtual-cavity model, because a fictitious sphere is


introduced as a trick for calculating the local field acting on a typical dipole in the

medium. An alternative, more elegant, derivation of the relationship (1.6) which does

not require introducing an imaginary sphere was described by Aspnes [37].

We further derive the Lorentz–Lorenz (or Clausius–Mossotti) relation for the di-

electric permittivity ǫ(1) and microscopic polarizability α. Let us assume for now that

the medium is lossless and dispersionless. We represent the dipole moment induced

in a typical molecule (or atom) of the medium as

p = αEloc. (1.7)

The macroscopic polarization of the material is given by the equation

P = N p, (1.8)

where N denotes molecular (or atomic) number density. Using Eqs. (1.6) through (1.8),

we find that the polarization and macroscopic field are related by

P = Nα

(

E +4π

3P

)

. (1.9)

We assume the polarization P to be linear in the average field:

P = χ(1)E, (1.10)


where χ(1) is the linear optical susceptibility of the medium. Substituting the expres-

sion (1.9) into (1.10), solving for χ(1), and eliminating the field E, we find that

χ(1) =Nα

1 − 4π

3Nα

. (1.11)

Expressing the left-hand side of (1.11) as χ(1) = (ǫ(1) − 1)/4π (ǫ(1) is the dielectric

permittivity of the medium), we obtain the well-known Lorentz–Lorenz (or Clausius–

Mossotti) relation

ǫ(1) − 1

ǫ(1) + 2=

4π

3Nα. (1.12)

Through rearrangement of Eq. (1.12), we can express the linear susceptibility as

χ(1) =ǫ(1) + 2

3Nα. (1.13)

Substituting the expression (1.13) into (1.10), then (1.10) into (1.8), and using the re-

lationship (1.7) between the local field and the dipole moment, we obtain the equation

relating the local field to the average field:

Eloc =ǫ(1) + 2

3E. (1.14)

The factor

L =ǫ(1) + 2

3(1.15)


is known in the literature as the Lorentz local-field correction factor. The expres-

sion (1.15) for the local-field correction factor is valid in the case of homogeneous

media, where all the particles (molecules or atoms) are of the same sort. It is also

valid in materials where the emitters enter as inclusions that do not influence the

correlation between the host molecules or atoms [12,20].

Onsager Model

A different macroscopic model for describing the local field in homogeneous media

had been developed by Onsager [38]. In his study, Onsager treats a molecule or atom

as being enclosed in a tiny real cavity in the medium. Then the field acting on the

molecule is divided into the cavity field, which would exist at the center of the real

cavity surrounding the molecule in the absence of the molecule, and the reaction field,

which corrects the cavity field for the polarization of the surrounding medium by the

dipole field of the molecule in the cavity. The resulting local field is given by

Eloc =3ǫ

2ǫ + 1E +

2(ǫ(1) − 1)

(2ǫ(1) + 1)a3p (1.16)

with the first and second terms expressing the cavity and reaction fields, respectively.

Here a is the cavity radius. Even though the Lorentz and Onsager models yield

different expressions for the local field, microscopic theories, developed in [12] and [39],

reconcile those two models, which appear to be two special cases of the more general

theories.


In reality, there is no need to apply a full microscopic treatment of local-field

effects in order to describe an experimental outcome. In most of the cases, one of

the two macroscopic theories works reasonably well. Onsager model is applicable to

polar liquids, while the Lorentz model is applicable to solids and non-polar liquids.

Both models can describe a guest-host system. The Lorentz model describes such

a system in cases when the guest’s molecule or atom replaces a molecule or atom

of the host with similar polarizability [12]. An example is neodymium-doped YAG

where neodymium guest ions replace yttrium ions in the crystalline structure. Both

neodymium and yttrium belong to the class of rare-earth metals, which implies that

they have similar properties. Onsager model is more suitable when the polarizability

of a guest is significantly different from that of the host molecules or atoms. Then the

guest not only forms a cavity in the host medium, but affects the local field outside

the cavity [38,40]. A good example of such a guest-host system is provided by liquid

solutions of fullerene C60 [40].

Real-Cavity Model

Real-cavity model is a simplified version of the Onsager model of local field. Like in

the Onsager model, within the real-cavity model one considers an emitter as being

inserted into a tiny cavity inside a dielectric medium [11]. The cavity is assumed

to have no material in it except for the emitting dipole under consideration. This

situation can be realized in the case when the polarizability of the emitter is low,


so that it does not impose strong changes on the local field outside the cavity. The

expression for the local field, given by the real-cavity model, has the form

Eloc =3ǫ(1)

2ǫ(1) + 1E, (1.17)

and the corresponding local-field correction factor is given by [11]

L =3ǫ(1)

2ǫ(1) + 1. (1.18)

Comparing Eqs. (1.16) and (1.17), it is easy to see that within the real-cavity model

one neglects the reaction field. Despite this approximation, real-cavity model de-

scribes many experimental outcomes reasonably well, which we show in Chapter 2.

1.3.2 Local Field and Cascading in Nonlinear Optics

Local-field effects lead to a modification of the optical properties of dense media, and,

consequently, serve as a source of interesting new nonlinear optical phenomena. For

instance, steady-state solutions to the local-field-corrected Maxwell–Bloch equations

indicate that it is possible to realize mirrorless optical bistability [41–44]. Also, an ad-

ditional inversion-dependent frequency shift appears. This frequency shift, called the

Lorentz red shift, was experimentally measured in the reflection spectrum of a dense

alkali vapor [45, 46]. The Lorentz red shift can cause a pulse to acquire a dynamic

chirp, which enables soliton formation at a very low level of atomic excitation [47,48].


In a collection of three-level atoms, local-field effects can lead to inversionless gain

and the enhancement of the absorptionless refractive index by more than two orders

of magnitude [49–52]. Successful experimental attempts to realize this enhancement

of the refractive index have been reported [53,54].

A phenomenological approach to treating local-field effects in nonlinear optics was

proposed by Bloembergen [55]. He found that the local-field-corrected second-order

nonlinear susceptibility scales as three powers of the local-field correction factor L. It

has been understood that Bloembergen’s result can be generalized to a higher-order

nonlinearity, and that the corresponding ith-order nonlinear susceptibility should

scale as Li+1 (see, for example, [56, 57]). In Chapter 3 we show theoretically that

Bloembergen’s approach, when consistently applied, actually leads to a much more

complicated form for the nonlinear susceptibility. This is due to the presence of a

cascaded nonlinear effect.

Cascading is a process in which a lower-order nonlinear susceptibility contributes

to higher-order nonlinear effects in a multi-step fashion; it has been a field of interest in

nonlinear optics for some time. Macroscopic cascading has a non-local nature, in that

the intermediate field generated by a lower-order nonlinearity propagates to contribute

to a higher-order nonlinear process by nonlinearly interacting with the fundamental

field [58–67]. Thus, it has been acknowledged that the experimentally-measured third-

order susceptibility can include contributions proportional to the square of the second-

order susceptibility [58–60]. On the other hand, it has also been shown that nonlinear


cascading is possible due to the local nature of the field acting on individual molecules

in the medium [60,68–73]. This local-field-induced “microscopic” cascading does not

require propagation, and has a purely local character.

The fact that local-field effects create cascaded contributions of the lowest or-

der hyperpolarizability γ(2)at to the third-order susceptibility was first demonstrated

by Bedeaux and Bloembergen [68]. They presented a general relationship between

macroscopic and microscopic nonlinear dielectric response, obtained neglecting the

pair correlation effect, which was later taken into account by Andrews et al. [72]. All

the studies conducted thus far have concentrated on treating the local cascading con-

tribution of γ(2)at to third-order nonlinear effects, which only arises if the constituent

molecules lack center of inversion symmetry. We show in Chapter 3 both theoreti-

cally and experimentally that the microscopic cascading effects can be significant in

higher-order nonlinearities, and are present in any system with the nonlinear response

higher than the lowest-level.

Together with the fundamental interest to the microscopic cascading effect, there

is a practical significance for its detailed investigation. Many studies in quantum

information science require materials that can respond to the simultaneous presence

of N photons, as in N-photon absorption. Usually, lower-order nonlinearities are

stronger than higher-order nonlinearities. Microscopic cascading allows to synthesize

higher-order nonlinearities out of lower-order nonlinearities by means of local-field

effects, and, therefore, has a potential in application for high-order nonlinear optical


materials.

1.4 Lorentz–Maxwell–Bloch Equations

Within the next two chapters, we will extensively use Maxwell–Bloch equations with

the Lorentz model used for accounting for local-field effects. We call this formalism

“Lorentz–Maxwell–Bloch Equations.” We find it informative to describe the formal-

ism in the current chapter.

A collection of two-level atoms with ground and excited states denoted respectively

by a and b, interacting with an optical field closely tuned to an atomic resonance of

the system, can be described by the Maxwell–Bloch equations [41,56]

σ =

(

i∆ − 1

T2

)

σ − 1

2iκEw (1.19a)

and

w = −w − weq

T1

+ i(κEσ∗ − κ∗E∗σ). (1.19b)

Here E(t) is the slowly varying amplitude of the macroscopic electric field E(t) =

E(t) exp(−iωt) + c. c., and the total (linear and nonlinear) polarization P (t) =

P (t) exp(−iωt) + c. c. involves

P (t) = Nµ∗σ(t), (1.20)


where N is the number density of atoms, µ is the dipole transition moment of the

two-level system from the ground to excited state, and σ(t) is the slowly-varying

amplitude of coherence σ(t), that is,

σ(t) = σ(t) exp(−iωt). (1.21)

In Eqs. (1.19), κ = 2µ/~, ∆ = ω−ωba is the detuning of the optical field frequency ω

from the atomic resonance frequency ωba, T1 and T2 are respectively the population

and coherence relaxation times, w is the population inversion, and weq is its equilib-

rium value. According to the prescription of Lorentz [5], the field that appears in

Eqs. (1.19) is actually the local field Eloc, given by Eq. (1.6). Substituting Eq. (1.6)

into the Maxwell–Bloch Eqs. (1.19), we obtain

σ =

(

i∆ + i∆Lw − 1

T2

)

σ − 1

2iκwE (1.22a)

and

w = −w − weq

T1

+ i(κEσ∗ − κ∗E∗σ). (1.22b)

The term ∆Lw entering the equation for σ introduces an inversion-dependent fre-

quency shift, which is a consequence of local-field effects; the quantity ∆L is called

Lorentz red shift and is given by

∆L = −4π

3

N |µ|2~

. (1.23)


The steady-state solutions to Eqs. (1.22) are

w =weq

1 +|E|2/|E0

s |21 + T 2

2 (∆ + ∆Lw)2

(1.24a)

and

σ =µ

~

wE

∆ + ∆Lw + i/T2

. (1.24b)

In (1.24a) we introduced saturation field strength E0s , defined as

|E0s |2 =

~2

4T1T2|µ|2. (1.25)

The total polarization can be expressed in terms of the total susceptibility χ,

including linear and nonlinear optical response, as P = χE. From (1.20), we find

that

χ =P

E=

Nµ∗σ

E. (1.26)

Substituting the steady-state solution for the coherence σ in the form (1.24b) into

Eq. (1.26), we obtain

χ =N |µ|2T2

~

w

T2(∆ + ∆Lw) + i. (1.27)

We use the results presented in this Section for describing laser gain properties of

composite optical materials in Chapter 2, and as an approach to treat the microscopic

cascading in high-order nonlinear response in Chapter 3.


1.5 Dye-Doped Cholesteric Liquid Crystal Lasers

Dye-doped cholesteric liquid crystals (CLCs) are self-assembling mirrorless distributed-

feedback low-threshold laser structures. The idea of lasing in CLCs was first proposed

by Goldberg and Schnur in 1973 [74]. Independently, Kukhtarev proposed the idea

and developed a theory of CLC distributed feedback lasers in 1978 [75]. First exper-

imental observation of lasing action in CLCs and a number of follow-up experiments

were performed by Ilchishin, et al. [76–78] in 1980, almost twenty years before the

nature of lasing in CLCs at the photonic band edge was explained by Kopp, et al. [79].

Taheri, et al. [80] reported the observation of laser action in CLC structures shortly

after Kopp. CLC lasers became a subject of a great interest over the past decade.

They combine tunability of dye lasers together with compactness and robustness of

semiconductor lasers. Because of their small size, tunability, and low-cost fabrication,

these lasers have a great potential to be used in medicine and LCD technology.

Cholesteric liquid crystal structures are produced by mixing a nematic liquid crys-

tal with a chiral additive that causes the nematically-ordered molecules to arrange

themselves into a helical structure. In the planes perpendicular to the helical axis of

the structure the molecules have a nematic-like order, aligning along some preferred

direction. This direction can be characterized by a unit vector called local director.

The local director rotates from plane to plane as we look along the helical axis of

the CLC structure. The distance along the helical axis needed for the director to

complete a full circle is called helical pitch P .


Alignment of the rod-shaped molecules along a preferred direction causes the CLC

structures to exhibit local birefringence. The refractive indices “seen” by the light

polarized along the local director and in perpendicular direction are called extraor-

dinary (ne) and ordinary (no), respectively. When circularly polarized light with the

same handedness as that of the helical structure propagates along the CLC helical

axis, it sees a periodic modulation of the refractive index and a selective reflection, if

the wavelength of the light is in a certain range. This range is defined by the pitch

of the CLC according to

∆λ = P∆n, (1.28)

where

∆n = ne − no. (1.29)

Thus, a CLC can be regarded as a 1D photonic crystal with the center of the photonic

band gap defined by the wavelength

λc = nP, (1.30)

where

n =no + ne

2(1.31)

is the average of the refractive indices of the CLC. Propagation of circularly polarized

light with the handedness opposite to that of the CLC is unaffected by the structure


and experiences merely the average refractive index (no + ne)/2. Here we call a

circularly polarized wave right-handed if its electric field vector appears to rotate

clockwise as the wave propagates towards an observer, and left-handed if its electric

field vector rotates counter-clockwise. This is a standard definition of handedness of

circularly polarized light [34].

If one dopes a CLC structure with an organic dye that has the emission spectrum

overlapping with the CLC photonic band gap, one can observe changes in photolumi-

nescence of the dye. The emission will be enhanced at the low- and high-frequency

photonic band edges and suppressed at the band gap for the circularly polarized com-

ponent with the same handedness as that of the CLC structure. The sharp rise of the

photoluminescence at the band edges is due to the fact that the density of states is

very high at these spectral ranges. The circularly polarized photoluminescence with

the opposite handedness does not experience any changes in the CLC.

A CLC host can serve as a resonator for a laser dye doped into it. The enhance-

ment of the dye photoluminescence at the band edges, caused by high density of

electromagnetic states, leads to easily achievable low-threshold lasing [79]. The two

lasing modes having the lowest threshold are situated at the band edges. The mode

on the low-frequency band edge is comprised of two circularly polarized counter-

propagating waves, resulting in a standing wave with the electric field vector aligned

along the local director (provided that no < ne) [81]. The mode at the high-frequency

band edge is similar to that at the low-frequency edge, except in this case the electric


field vector is perpendicular to the local director. Depending on the mutual align-

ment of the dye transition dipole moment and the local director, lasing oscillations

can occur at the low-frequency, high-frequency, or at both band edges [81,82].

There are several ways to characterize the orientation of the dye dipole moment

with respect to the local director [81–84]. The most widely used characteristic is the

dye emission order parameter Sem, given by the expression

Sem =I‖ − I⊥I‖ + 2I⊥

, (1.32)

where I‖ is the fluorescence intensity of the nematic liquid crystal phase doped with

the dye, measured for the radiation with the electric field parallel to the director,

and I⊥ is the fluorescence intensity for the radiation with the electric field polarized

perpendicular to the director. Obviously, the case Sem = 1 corresponds to a perfect

alignment of the dye dipole moment along the liquid crystal director, the case Sem =

−1/2 corresponds to a perfect alignment of the dye dipole moment perpendicular to

the director, and the case Sem = 0 corresponds to an isotropic orientation. In our

work reported in Chapter 4, we use the characteristic given by Eq. (1.32) to describe

the dye dipole moment alignment.


1.6 Planar Chiral Structures

A planar object is said to be chiral if it cannot be brought in congruence with its

mirror image unless it is lifted from the plane [85]. Here we will distinguish between

two forms of planar chirality: “molecular” and “structural.” The molecular chirality

is a property of an individual particle to lack mirror symmetry (an example of such

a particle is a gammadion), while the structural chirality is a property of the entire

structure. For example, an array of particles possessing molecular chirality is, in

whole, structurally chiral. Even a regular array of achiral particles can possess struc-

tural chirality, if the particles are arranged in such a way that the whole structure

lacks mirror symmetry.

The observation of the chirality-sensitive diffraction in planar arrays of chiral gold

nanoparticles by Papakostas et al. [85] has attracted a lot of attention to the interac-

tion of light with artificial planar chiral structures. Because of their ability to change

the polarization state of diffracted [85–89] and transmitted [90, 91] light, artificial

planar chiral structures have a great potential of being used as polarization control

devices. It has been shown that such structures can demonstrate specific optical

activity three orders of magnitude stronger than that produced by quartz [90]. Polar-

ization conversion and focusing of circularly-polarized light on transmission through

a gammadion-shaped hole and arrays of chiral particles have been demonstrated both

numerically [92–94] and experimentally [95]. The issue of temporal reversibility and

spatial reciprocity in planar chiral structures has been a subject of a great contro-


versy [89,90,95–97]. Another property contributing to the practical importance of the

artificial planar structures is their ability to display unusual nonlinear optical proper-

ties because of their broken symmetry imposed by the chirality itself or by small-scale

irregularities. In particular, it has been shown by Canfield et al. [98–102] that the

smallest manufacturing defects in the arrays of gold L-shaped nanoparticles can re-

sult in drastic enhancement of second-harmonic generation, and even in appearance

of forbidden tensor components in second-harmonic output.

So far, in all the experimental investigations of the polarization changes of light

in transmission [90, 91] and diffraction [85–89] by planar chiral structures, the focus

was made on studying the structures with chiral individual particles. To the best of

our knowledge, there are no experimental studies of polarization changes in artificial

planar structures with achiral particles arranged in a chiral fashion.

In Chapter 5 we report the observation of polarization changes in diffraction from

periodic patterns with pure structural chirality. The polarization effects were found

to be large, in fact, as large as those observed from reference patterns consisting

of chiral particles. The results suggest that structural and molecular chirality are

indistinguishable and cannot be separated from each other in diffraction experiments.

Indeed, both cases lead to the lack of mirror (and point inversion) symmetry in the

experimental geometry, making the whole setup “chiral.” The role of the chiral setup

is further emphasized by the fact that our results can be explained by considering

scattering from individual particles.


Here I have presented an overview of all projects that I have been working on

during my Ph. D. research, and the basic concepts that these projects rely on. More

details and my contributions in particular can be found in separate chapters devoted

to the projects.

Chapter 2

Composite Laser Materials

2.1 Introduction

Nanocomposite materials can differ significantly in their optical properties from bulk

constituent materials [9, 10, 25, 29, 103], and their growing importance in photonics

calls for a better understanding and characterization of the role of local-field and

other effects that influence their optical properties.

There are numerous works describing the nonlinear optical properties of composite

materials [25–28]. However, there has not been a systematic study yet of their laser

properties. In this chapter we present a study of the influence of the local-field effects

on the laser properties of composite materials. We first present a general picture

of the idea of composite laser in Section 2.2, and show the significance of the local-

field effects in modifying laser properties. In Section 2.3 we present our experimental

27

CHAPTER 2. COMPOSITE LASER MATERIALS 28

study of the radiative lifetime of Maxwell Garnett-type composite materials based

on liquid suspensions of Nd:YAG nanoparticles. We show that local-field effects can

significantly vary the radiative lifetime of Nd:YAG. Local-field effects can come into

play differently in different composite geometries, and separate theoretical studies of

the laser properties of various composite geometries are needed. In Section 2.4 we

present such studies for Maxwell Garnett and layered composite geometries. Finally,

in Section 2.5 we summarize our theoretical and experimental studies of composite

laser materials.

2.2 The Idea of Composite Lasers

In this section, we limit ourselves to a simple treatment of a composite material as

a uniform medium characterized by a dielectric constant ǫeff . At the present level of

approximation, the Lorentz model can be used to describe the effects of the local field

on the laser properties of the medium. We recall for convenience the expression for

the Lorentz local field from Eq. (1.6),

Eloc = E +4π

3P, (2.1)

and the Lorentz local-field correction factor from Eq. (1.15),

L =ǫ(1) + 2

3, (2.2)


relating the local and average fields by Eq. (1.5). As we pointed out in Section 1.2,

composite materials can be treated as effective media, as the sizes of the particles

of the constituent components are much smaller than the optical wavelength. Under

this condition we can consider a composite material as an effective medium charac-

terized by an effective (average) dielectric constant ǫeff . Thus, at the present level

of approximation, we can use Eq. (2.2) for the local-field correction factor with the

effective dielectric constant in place of ǫ(1).

2.2.1 Influence of Local-Field Effects on Laser Properties of

Dielectric Materials

In this subsection we describe the modification of laser properties, such as the ra-

diative lifetime, the small-signal gain coefficient, and the saturation intensity by the

local-field effects. We undertake our analysis based on a simple argument of the

validity of the Lorentz model for treating the local-field effects.

Radiative Lifetime

The radiative lifetime τ of emitters in a dielectric material depends on the dielectric

constant of the material. It is inversely proportional to the Einstein A coefficient,

which, in turn, can be expressed through Fermi’s golden rule as

A =1

τ=

2π

~|V12(ω0)|2ρ(ω0). (2.3)


Here V12(ω0) is the energy of interaction between the emitter and the electric field

in the medium, and ρ(ω0) is the density of states at the emission frequency ω0. In a

medium with refractive index neff in which the local-field effects are significant, the

interaction energy scales as

V12, loc ∝L√neff

. (2.4)

The factor L enters the expression for the local-field-corrected interaction energy

V12, loc(ω0) because the local field acting on an individual emitter differs from the

macroscopic average field. The factor√

neff in the denominator of (2.4) comes from

the mode normalization and thus appears in the expression for the electromagnetic

energy density in a dielectric medium [13]. The density of states in the medium is

proportional to the square of the effective refractive index:

ρ(ω0) ∝ n2eff . (2.5)

Using Eqs. (2.3) through (2.5), we can conclude that the local-field-corrected spon-

taneous emission rate Aloc in the medium with refractive index neff is related to the

spontaneous emission rate in the medium of unit refractive index (we call it Avac) as

Aloc = neff |L|2Avac. (2.6)

The relation (2.6) has been shown to hold also when the effect of dispersion is included

in V12, loc and in the density of states [13]. The corresponding relation for the local-


field-corrected radiative lifetime τloc in terms of the “vacuum” lifetime τvac takes form

τloc =τvac

neff |L|2. (2.7)

Here and in all later sections of this chapter we assign the “vac” subscript to the

variables denoting quantities in a medium with the same chemical environment as that

of the dielectric medium under consideration, but with the refractive index equal to

unity. The variables marked with the “loc” subscript denote the local-field-corrected

quantities.

Small-Signal Gain

Most laser gain media can be modeled as collections of two-level atoms, regardless

of what the actual level diagram of the active medium is, because in most cases

the nonradiative transitions are much faster than the radiative transition from the

upper laser level. We use the two-level-atom model to derive the expression for the

local-field-corrected small-signal gain.

We start from the driven wave equation

−∇2E +1

c2

∂2E

∂t2= −4π

c2

∂2P

∂t2. (2.8)


(We limit ourselves to considering scalar fields for simplicity.) Here

E(t) = E(z)e−iωt + c. c. = A0ei(kz+ωt) + c. c. (2.9)

is the average electric field. In the general case of a lossy or amplifying medium, the

parameter k is complex: k = k + iα0/2. The real part of k is the wave number,

Re(

k)

≡ k =ωneff

c, (2.10)

where neff is the effective refractive index of the medium, while the imaginary part of

k characterizes amplification or attenuation in the medium, according to

Im(

k)

=1

2α0 = −1

2g0. (2.11)

Here α0 and g0 are the small-signal intensity absorption and gain coefficients, respec-

tively. We seek the solution for the local-field-corrected small-signal gain coefficient,

which we denote as g0, loc.

We take the polarization P entering the wave equation (2.8) to be linear:

P = χ(1)locE, (2.12)

where χ(1)loc is the local-field-corrected linear susceptibility. Using Eq. (1.27) for the

susceptibility of a collection of two-level atoms and neglecting the nonlinear depen-


dence on the electric field, we obtain [56]

χ(1)loc = −cα0, vac(∆)

4πωba

L(T2∆ − i), (2.13)

where ωba is the frequency of the atomic transition, ∆ = ω−ωba is the detuning of the

optical field with respect to the atomic transition frequency, and T2 is the coherence

relaxation time. The “vacuum” absorption α0, vac(∆) experienced by a weak optical

wave detuned from the resonance is given by [56]

α0, vac(∆) = −4πωba

c

Nweq|µba|2T2

~(T 22 ∆2 + 1)

, (2.14)

where weq is the equilibrium value of the population inversion, N is the atomic number

density, and µba is the transition dipole moment of the two-level atom.

The local-field correction factor L, given by Eq. (2.2) as L = (ǫeff + 2)/3, can be

shown to take the form

L =T2∆ + i

T2(∆ − ∆L) + i, (2.15)

if one chooses to express ǫeff in terms of atomic parameters as [5, 45]

ǫeff = 1 − freλ0cN

∆ − ∆L + i/T2

. (2.16)

Here ∆L = −(freλ0cN)/3 is the Lorentz red shift [45] (f is the oscillator strength of

the atomic transition, re = e2/mc2 is the classical electron radius, and λ0 is vacuum


transition wavelength). The Lorentz red shift appears in the expression for ǫeff as a

consequence of the local-field effects and leads to L 6= 1 (note that if ∆L = 0, then

L = 1, and there are no local-field effects).

Substituting the electric field given by Eqs. (2.9) through (2.11) and the polar-

ization in the form of Eq. (2.12) into the wave equation (2.8), then taking the time

and space derivatives and dropping the electric field that appears as a multiplicative

factor on the both sides of the resulting equation, we end up with the equation

g0, loc = −4πk

n2eff

Im[

χ(1)loc

]

(2.17)

for the gain coefficient. Substituting the factor L given by (2.15) into the expres-

sion (2.13) for the linear susceptibility, and then (2.13) into (2.17), we arrive at the

expression

g0, loc(∆) = −α0, vac(∆)

neff

ω

ωba

|L|2 =g0, vac(∆)

neff

ω

ωba

|L|2 (2.18)

for the local-field-corrected small-signal gain in terms of the “vacuum” absorption

coefficient α0, vac(∆), or “vacuum” gain coefficient g0, vac(∆) (i. e., the absorption or

gain coefficient in a medium with the unit refractive index). Assuming that the

optical wave is in resonance with the atomic transition (∆ = 0), we find the following

expression for the local-field-corrected gain coefficient:

g0, loc(0) =g0, vac(0)

neff

|L|2. (2.19)


It is in agreement with that obtained by Milonni [13].

Saturation intensity

In this subsection we use the following convention regarding the way we denote dif-

ferent variables. We assign the “vac” subscript to the variables describing quantities

in a medium with unit refractive index, no subscripts to variables denoting non-local-

field-corrected quantities in a medium with the refractive index neff , and the “loc”

subscript to variables denoting local-field-corrected quantities.

The gain g for a homogeneous atomic transition saturates with increasing signal

intensity I = (neffc|E|2)/(2π) according to [104]

g =g0

1 + I/Is

. (2.20)

The saturation intensity is the intensity that reduces the small-signal gain coefficient

to a half of its value; it is given in terms of the atomic transition parameters by [56]

Is =cneff

2π

~2

4|µba|2T1T2

. (2.21)

Here T1 is the population relaxation time.

To account for local-field effects in Eq. (2.20) for saturated gain coefficient, one

has to substitute the local-field-corrected counterparts of the quantities entering the

equation. We have obtained an equation for the local-field-corrected small-signal gain


coefficient g0, loc (see Eq. (2.19)). The local-field-corrected intensity can be written as

Iloc =neffc

2π|Eloc|2 =

neffc

2π|L|2|E|2 = |L|2I. (2.22)

The expression (2.21) for the saturation intensity contains two quantities that may

be affected by local-field effects, namely: T1 and T2. We assume that T2 does not

depend on the local-field correction factor, as would be true for many line broadening

mechanisms. Thus, we can simply retain the “vacuum” value of T2 in the equation for

the saturation intensity. T1 is the lifetime of the upper laser level, which we assume

to be purely radiative. Thus, the result (2.7) obtained earlier applies:

T1, loc =T1, vac

neff |L|2. (2.23)

Making use of Eq. (2.23) and the assumption that T2 does not introduce local-field

correction to the expression for the saturation intensity, we find that

I ′s =

cneff

2π

~2

4|µba|2T1, locT2, vac

=cneff

2π

~2neff |L|2

4|µba|2T1, vacT2, vac

. (2.24)

In Eq. (2.24) we denote the saturation intensity I ′s in order to discriminate it from

both the non-local-field-corrected saturation intensity Is and the local-field-corrected

saturation intensity Is, loc, as we did not account for all the local-field corrections

affecting the saturation intensity yet.


The local-field-corrected saturated gain coefficient can be written as

gloc =g0, loc

1 +Iloc

I ′s

(2.25)

in terms of the quantities given by Eqs. (2.19), (2.22), and (2.24). We define the

local-field-corrected saturation intensity so that, if written in terms of it, Eq. (2.25)

takes the form

gloc =g0, loc

1 +I

Is, loc

. (2.26)

Making use of Eqs. (2.24) through (2.26), we express the local-field-corrected satura-

tion intensity in terms of the “vacuum” saturation intensity as

Is, loc = n2effIs, vac. (2.27)

2.2.2 Analysis

The basic operation of lasers can be characterized most simply in terms of the upper-

level spontaneous emission lifetime τ , the laser gain coefficient g, and the gain sat-

uration intensity Is. All three of these parameters can be controlled through use of

a composite material geometry. In the simplest formulation of local-field effects, we

can assume that these laser parameters depend only on the effective value neff of the

refractive index of the host material. We showed in Subsection 2.2.1 that the basic


laser parameters scale with the effective refractive index according to

Aloc = neff |L|2Avac, (2.28a)

g0, loc =|L|2neff

g0, vac, (2.28b)

and

Is, loc = n2effIs, vac. (2.28c)

The quantities marked with “vac” denote the values of the parameters in a medium

with the same chemical environment, but with the refractive index equal to unity.

Treating the composite laser gain medium as an effective medium, and assuming

that the amplification (and loss) at the laser transition frequency is small enough to

neglect the imaginary part of the effective dielectric constant ǫeff , we can express the

local-field correction factor, given by Eq. (2.2), as

L =n2

eff + 2

3. (2.29)

Making use of Eq. (2.29) for the factor L, we plot the local-field-corrected basic laser

parameters, given by (2.28), in Figure 2.1. We choose the range of refractive indices

available in dielectric composite materials. Clearly, significant control over the laser

parameters is available through use of a composite geometry.

Control of the three laser parameters is crucial for the development of laser systems


0

10

20

30

40

50

1 1.5 2 2.5 3 3.5 4

Enh

ance

men

t

neff

A

g0

Is

Figure 2.1: Variation of the principal parameters that control the basic operation

of a laser with the effective refractive index of the composite material.

for the following reasons.

1. The upper state lifetime controls how large the pumping rate of the laser needs

to be in order to establish a population inversion.

2. The gain coefficient determines the laser threshold condition. The gain needs

to be large enough for the laser to reach threshold, but it is not desirable for

the gain to be too large because excessive gain can lead to the development of

parasitic effects such as amplified spontaneous emission.

3. The saturation intensity (and its related energy quantity, the saturation fluence)

determine the output power of a laser, from the point of view that the output

power is determined by the condition that the saturated round-trip gain must

equal the round trip loss. In practice, the output intensity of most lasers is

typically a factor of several times the saturation intensity.


Our assumption that local-field effects in composite laser gain media can be accounted

for by using the Lorentz model with the effective refractive index entering the expres-

sion for the factor L is good for conceptual understanding of how the local-field

effects help one to manipulate the laser parameters. We refer readers to Section 2.4

for a more precise and detailed analysis of Maxwell Garnett and layered composite

geometries.

The theoretical analysis presented in this section has been published in the Journal

of the Optical Society of America B [105].

2.3 Influence of Local-Field Effects on the Radia-

tive Lifetime of Liquid Suspensions of Nd:YAG

Nanoparticles

It is well known that the rate of spontaneous emission depends in general on the

environment of the emitter. The case in which an excited atom is inside a cavity or

near a reflecting surface has been studied extensively both theoretically and exper-

imentally [106]. Radiative processes in bulk dielectric media appear to be less well

understood at a fundamental level, and in particular there has been much current

interest in the effects of local fields on spontaneous emission [12].

Recent experimental work includes measurements of the radiative lifetimes of Eu3+

complexes in liquids [16], supercritical gases [17], and glass [18,19]. In this section we


present the results of measurements of radiative lifetimes of Nd:YAG nanopowders

in different liquids. Unlike previous experiments in which the Eu3+ is embedded in a

ligand cage, surface effects must be considered in the case of nanoparticles of the type

used in our work. However, despite these complications, the experiments described

in this section allow some important conclusions to be drawn about local-field effects.

Also, composite materials of the sort studied here may prove useful in the development

of photonic devices, and thus an understanding of their optical properties is especially

important.

We recall from Section 2.2 that the radiative lifetime in a dielectric medium can

be related to its vacuum counterpart as

τloc =τvac

neff |L|2(2.30)

(see also Eq. (2.7)). Existing theoretical models predict different expressions for

the local-field correction factor L. Two models, describing most of the experimen-

tal outcomes, are the virtual-cavity model (Lorentz model) [37], and the real-cavity

model [11] (see Section 1.3.1 for a more detailed overview). Within this section we

will refer to Lorentz model as “virtual-cavity model” in order to emphasize its dif-

ference from the real-cavity model. Using the virtual-cavity approach, one arrives at

the expression (2.29) for the local-field correction factor [5, 6]. Here we rewrite this

expression as

L =n2

eff + 2

3(2.31)


in terms of the effective refractive index of the composite dielectric medium. We

also rewrite the expression for the local-field correction factor following from the

real-cavity model (1.18) as

L =3n2

eff

2n2eff + 1

. (2.32)

The expressions (2.31) and (2.32) for the local-field correction factor are very dif-

ferent, and lead to different results for the radiative lifetimes in a dielectric medium.

In fact, they describe different physical situations [12, 20]. The Lorentz model ap-

plies only to homogeneous dielectric media [45], while the real-cavity model can also

describe the local-field effects in a medium where the emitters enter in the form

of inclusions, or dopants, creating tiny real cavities. Examples where the latter

model applies well are dye molecules dissolved in water droplets suspended in dif-

ferent liquids [21, 22], Eu3+ organic complexes suspended in liquids [16] and super-

critical gas [17], liquid suspensions of quantum dots [23] (with the interpretation of

the results given in Ref. [20]), and Eu3+ ions embedded into a binary glass system

xPbO − (1 − x)B2O3 [18, 19].

To the best of our knowledge, there is only one reported experiment on the inves-

tigation of local-field effects on the radiative lifetime of liquid suspensions of nanopar-

ticles [107]. 1 Based on the theoretical analysis conducted by P. de Vries and A. La-

gendijk [12], and on numerous examples of similar experiments [16, 17, 21–23], one

would expect the radiative lifetime of liquid suspensions of nanoparticles to obey the

1The composites containing Eu3+ embedded in a ligand cage [16,17] are fundamentally differentfrom the sort of nanocomposite material considered in [107] and in the present work.


real-cavity model. Indeed, when placed in a liquid, the nanoparticles create real cav-

ities, replacing some volume of the liquid. It is therefore somewhat surprising that

the experimental measurements of the radiative lifetime of Eu3+ : Y2O3 nanoparticles

in liquids, reported in Ref. [107], obeyed the virtual-cavity model. We believe that

more experimental studies are needed for a better understanding of local-field effects

in liquid suspensions of nanoparticles.

Measurements of the fluorescence lifetime in suspensions of neodymium-doped

yttrium-aluminum garnet (Nd:YAG) particles as a function of the refractive index

of the surrounding medium were reported in Ref. [108], but not under conditions

allowing a study of the influence of local-field effects, which require the particles and

the distances between the particles to be much less than the optical wavelength. The

observed changes in the fluorescence lifetimes of the Nd:YAG particles were purely the

effect of the change in the refractive index on the density of states. The average size

of the particles used by the authors of Ref. [108] was on the order of a wavelength of

light (several hundreds on nanometers), which made the suspensions of such particles

unsuitable for measurements of local-field effects due to the host liquid.

In this section we present the results of measurements of the radiative lifetime

of Nd3+:YAG nanoparticles dispersed in different liquids. Our goal is to measure

the change in the radiative lifetime of emitters 2 caused by local-field effects and to

establish which model for the local-field correction factor works best for our case.

2In our work we consider the Nd3+:YAG nanoparticles to be the emitters in our compositematerials.


2.3.1 Sample Preparation and Experiment

The Nd:YAG nanopowder used in our experiments was manufactured by TAL Mate-

rials company. According to the SEM photograph of the nanopowder (see Fig. 2.2),

the average particle diameter was around 20 nm. The neodymium concentration was

chosen to be 0.9 at. %, which is the standard value for Nd:YAG laser rods. Because

of aggregation it was necessary to use appropriate surfactants to obtain good liquid

suspensions.

Figure 2.2: SEM picture of Nd:YAG nanopowder. The average particle size is 20

nm.

We used thirteen different organic and inorganic liquids for suspending the nanopar-

ticles. The liquids and the corresponding values of their refractive indices are shown

in Fig. 2.3. We used the Tween 80 surfactant for water and the aqueous immersion

fluid manufactured by Cargille Laboratories, and 12-hydroxystearic acid for oil-based

organic liquids (such as carbon tetrachloride and toluene). The aqueous suspensions

were sonicated for five minutes, and the resultant dispersions were very stable; the

nanoparticles remained in suspension for more than a month. A magnetic stirrer was


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��

��

!��

"#�$ �$��%�

�&'(��

) �$*�� +��

�,��&-��

..��

/ �� *��

/��(��

)* ��

01(��(#��22� #��(��%��

.3��

Figure 2.3: Liquids that we used for suspending Nd:YAG nanoparticles and the

values of their refractive indices.

used to dissolve the surfactants and to suspend the nanoparticles in organic solvents.

The most stable organic suspensions were achieved with alcohols, while the oil-based

suspensions were only good enough for quick lifetime measurements and sedimented

shortly after the measurements were done. The Nd:YAG volume fraction in all the

samples was 0.11 vol. %.

As the volume fraction fYAG of the nanoparticles in our samples is very low, we

can treat our suspensions as Maxwell-Garnett-type composite materials, and use the

relation (1.1) to calculate the effective refractive index neff , substituting the refractive

index of the liquids in place of the host refractive index, n2h = ǫh, and the YAG

refractive index in place of that of the inclusions, n2i = ǫi. The effective refractive

indices of our samples are very close to the refractive indices of the liquids.

We used a Spectra Physics femtosecond laser system to optically excite neodymium


ions in the Nd:YAG nanoparticle suspensions. The scheme of the experimental setup

is represented in Fig. 2.4. The radiation was generated by a mode-locked Tsunami

T s unami (800 nm, 1 MHz)

Mil lennia (532 nm)S pit f i r e

(800 nm, 250 Hz) f = 100 mm

f = 50 mm

Nd:YAG suspensions

Narrow-band filter for 1064 nm

InGaAs detector

Digital oscilloscope

f = 50 mm

Figure 2.4: Experimental setup for measuring the radiative lifetime in liquid

suspensions of Nd:YAG nanoparticles.

Ti:sapphire laser, providing 100-fs pulses with 800 nm central wavelength and an

initial repetition rate of 80 MHz. The Tsunami laser output was sent to a Spitfire

regenerative amplifier, and the repetition rate of the amplifier output was adjusted

to 250 Hz, in order to provide enough time between successive pump pulses. The

duration of the pulses exiting the regenerative amplifier was around 120 fs, and the

pulse energy was close to 1 mJ. The pump radiation was focused into a cell containing

the suspension by a lens with a focal length of 100 mm, and the fluorescence from the

Nd:YAG nanoparticles was collected by a 50 mm × 50 mm condenser in a perpen-

dicular geometry (see Fig. 2.4). A Thorlabs InGaAs detector and a Tectronix digital

oscilloscope were used to observe and record the fluorescence decay curves. A narrow-

band 10 nm FWHM filter with the central wavelength of 1064 nm, together with an

additional long-pass filter, were placed in front of the detector to block scattered


pump radiation.

A typical time trace showing the fluorescence decay dynamics is shown in Fig. 2.5.

In all our experiments we observed a non-exponential decay, and our data can be fitted

Figure 2.5: Typical fluorescence decay in the Nd:YAG nanopowder.

well with a sum of two exponentials with a 4:1 ratio of the slower to the faster decay

times. All the slower decay exponentials have fluorescence decay times longer than

the typical value 230 µs for bulk Nd:YAG, while all the faster decay exponentials

have decay times shorter than that of bulk Nd:YAG. We expect the fluorescence

decay times in our Nd:YAG nanopowder suspensions to be longer than that in a bulk

Nd:YAG crystal, because the effective refractive indices of our liquid suspensions

(1.32–1.63) are smaller than the refractive index of a bulk Nd:YAG (1.82). This is

one of the reasons why we chose the longer radiative decay time to be compared to

different theories describing the local-field effects. Other reasons will be evident from

our further analysis.


We performed a series of experiments to determine the origin of the shorter flu-

orescence decay component. Fluorescence decay dynamics similar to that shown in

Figure 3 was previously observed in a bulk Nd:YAG crystal as a consequence of ampli-

fied spontaneous emission [109]. Measuring the fluorescence decay at different pump

energies, we did not observe any variations in the two lifetimes and their correspond-

ing amplitudes. We also observed no changes when we varied the geometry of the

experiment. All this indicates that the amplified spontaneous emission is not the

reason for the bi-exponential fluorescence decay dynamics in our experiments.

Another possible reason for the faster exponential in the fluorescence decay in our

samples could be the contribution from ions sitting on the surfaces of the nanopar-

ticles. In order to check this hypothesis, we measured the fluorescence lifetimes not

only for the Nd:YAG nanopowder, but also for an Nd:YAG micropowder with a

micrometer-scale particle size. We obtained the micropowder by crushing an Nd:YAG

laser rod and grinding the pieces in a ball mill. The resultant decay dynamics dis-

played a bi-exponential character in both powders. The shorter decay time (130 µs)

was the same for both powders (within the error of our measurements), and the

longer decay time was around 600 µs for the nanopowder and more than two times

shorter (270 µs) for the micropowder. Furthermore, the relative contribution of the

faster-decay exponential was much higher for the nanopowder, compared to the mi-

cropowder. The fact that the shorter lifetime is the same for both powders, and that

the faster-exponential amplitude is relatively higher for the nanopowder, suggests


that the faster exponential in the fluorescence decay is due to the contribution from

the Nd3+ ions sitting on the surfaces of the particles. Indeed, the structures of the

surfaces of the nanoparticles and microparticles should be similar, while the relative

surface area of the nanopowder is much larger than that of the micropowder. The

crystal lattice surrounding of the surface ions is distorted, leading to the variations

in their electric dipole moments and resulting in much shorter relative decay times.

As surface effects are not the topic of our present research, we will concentrate on the

analysis of the slower radiative decay component.

2.3.2 Data Analysis

The fluorescence lifetimes for the various host liquids (shown as points in Fig. 2.6)

are obtained by fitting the time evolution of the fluorescence decay to the sum of two

exponentials and taking the longer decay time as the relevant time for the reasons

explained above. As the results of the fitting procedure were somewhat sensitive

to the range of time values used in the fitting procedure, we repeated the fit for

several different time ranges for each data point. The data points shown in Figure 2.6

represent averages of the results of these fits, and the error bars represent the standard

deviations from the mean values.

The radiative lifetime of the nanoparticles can be expressed as a function of the


effective refractive index of their liquid suspensions by

τrad =τ

(vac)rad

neff

(

n2eff + 2

3

)2 (2.33)

in the case when the local-field effects obey the Lorentz model, and by

τrad =τ

(vac)rad

neff

(

3n2eff

2n2eff + 1

)2 (2.34)

when the local-field effects obey the real-cavity model. Expressions (2.33) and (2.34)

can be obtained by substituting Eqs. (2.31) and (2.32), respectively, into Eq. (2.30).

In the absence of local-field effects, the radiative lifetime can be expressed in terms

of the effective refractive index of the suspension by the simple relation

τrad =τ

(vac)rad

neff

. (2.35)

Because we do not know its exact value, we will take the vacuum radiative lifetime

τ(vac)rad as an adjustable parameter for fitting our experimental data to different models

describing the local-field effects .

Generally, the measured decay time is not purely radiative and can be expressed

in terms of the radiative and non-radiative decay times as

1

τmeasured

=1

τrad

+1

τnonrad

. (2.36)


It is commonly assumed that the nonradiative decay time A−1nonrad does not depend on

the refractive index of the surrounding material [16,22] and can be roughly expressed

in terms of the radiative lifetime of ions in vacuum using the relation

η =A

(vac)rad

A(vac)rad + Anonrad

(2.37)

for the quantum yield η of the material. Quantum yield is the fraction of the energy

decaying through the radiative channel, so the measured lifetime is purely radiative

only in the case when the quantum yield of the material is close to unity.

We were unable to obtain reliable measurements of the absolute quantum yield

of our Nd:YAG nanoparticles, in part because of the strong scattering that these

particles produce. Instead, we have made use of published values of the quantum

yield from bulk samples. Reported values of the quantum yield range from 0.48 (see

Ref. [110]) to 0.995 (see Ref. [111]), with 0.6 being the value most often reported [112–

114]. It appears that the quantum yield for a given sample depends sensitively on Nd

concentration and on environmental issues. For this reason, we have fitted our data

to three different models, given by the equations (2.33)–(2.35), using the quantum

yield as an additional adjustable parameter.

We first fitted our experimental data to Eq. (2.33), which corresponds to the

Lorentz model of local field, and obtained the best least square fit for the quantum

yield value η ≈ 0.17 ± 0.02. This value of quantum yield does not fall into the

range of the typical measured values of the Nd:YAG quantum yield. This suggests


that the Lorentz (virtual-cavity) model is not likely to describe our experimental

outcome properly. Next we used Eq. (2.34) and obtained the best least-square fit

of our experimental data for the quantum yield value η ≈ 0.43 ± 0.07. In this case,

the value of the quantum yield falls into the range 0.48 ≤ η ≤ 0.995 within the

error, from which we conclude that the real-cavity model gives a good description to

our experimental data. Finally, we made an attempt to fit our experimental data to

Eq. (2.35), which corresponds to the assumption that there are no local-field effects.

The value of the quantum yield resulting in the best least-square fit is η ≈ 1.13±0.18,

which falls into the range of the measured quantum yield values within the error.

The value of the quantum yield giving the best least-square fit of our data with

the real-cavity model coincides with the low limit of the measured Nd:YAG quantum

yield, whereas such value for the no-local-field-effects model corresponds to the top

limit. In order to have an additional argument in favor of one of the two models that

give satisfactory descriptions to our experimental data, we present our experimental

data together with the results of the least-square fits for the two limiting values of

reported quantum yields, namely 1 and 0.48 in Figs. 2.6(a) and 2.6(b), respectively.

Through visual inspection of these results, one can immediately rule out the virtual-

cavity model. Under the assumption η = 1, both the real-cavity and the no-local-field-

effects models agree reasonably well the experimental data, with no-local-field-effects

model providing a slightly better fit [see Fig. 2.6(a)]. For the other limiting case, with

the assumption that η = 0.48 [see Fig. 2.6(b)], the real-cavity model gives the best


250

300

350

400

450

500

1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

Lif

etim

e (µ

s)

Effective refractive index

(a)

No local field effects

Real cavity

Virtual cavity 250

300

350

400

450

500

1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

Lif

etim

e (µ

s)

Effective refractive index

(b)

No local field effects

Real cavity

Virtual cavity

Figure 2.6: Experimentally measured radiative lifetimes of the Nd:YAG-

nanopowder suspensions (points with error bars), and the best least-squares ts

with various models (lines) under the assumption that the quantum yield of the

nanopowder is (a) 1 and (b) 0.48.

fit, which agrees with our expectations based on the theoretical analysis performed in

Ref. [12]. The no-local-field-effects model lies pretty close to our experimental data,

too. However, there is no theoretical justification for assuming the validity of the

no-local-field-effects model, as the physical properties of our samples were such that

local field effects should have been present. That is, the sizes of the particles were

more than 30 times smaller than the wavelength of light. Also the dispersions were

stable in most of the samples, which indicates that even if some particle aggregation

had occurred the aggregates were still smaller than the wavelength of light.

The experimental work reported in this section has been published in the Journal

of the Optical Society of America B [24].


2.4 Optimization of Laser Gain Properties for Lay-

ered and Maxwell Garnett Composite Geome-

tries

In Section 2.2 we show that the basic laser properties, such as the radiative lifetime

of the upper laser level, small-signal gain coefficient, and saturation intensity can be

controlled independently by means of local-field effects [105]. There we made an ap-

proximation, treating a composite material of any geometry as a quasi-homogeneous

medium characterized by an effective refractive index, with the local-field effects ac-

counted for using the Lorentz model. The Lorentz model of local field has been shown

to be applicable to homogeneous media [12]. However, local-field effects can mani-

fest themselves differently in composite materials, and separate theoretical models

for describing the laser properties of different composite geometries [10] are needed.

In this section, we present such models for layered and Maxwell Garnett composite

geometries. One can follow the recipe that we give in the current section to assess

the laser properties of more complex composite systems as well.

In Section 1.3.1 we presented an overview of the Lorentz and Onsager models,

which are used for describing local-field effects in homogeneous (non-composite) me-

dia. In Subsection 2.4.1 of the current section we present the modifications to the

Lorentz model for the case in which only a fraction of the atoms or molecules com-

prising a medium is in resonance with the applied optical field. In Subsection 2.4.2


we consider composite materials of layered geometry under the assumption that one

of the constituents contains molecules or atoms that are in resonance with the optical

field. We derive the complex susceptibility for such a system, from which one can de-

duce the absorption and gain coefficients. In the process of the derivation, we obtain

a modified local-field-induced frequency shift, similar to the well-known Lorentz red

shift [45] in “pure” systems with only one type of molecules or atoms. We perform

similar calculations for the Maxwell Garnett composite geometry in Subsection 2.4.3.

In Subsection 2.4.4 we analyze the results obtained in the previous subsections, con-

sidering two physical systems corresponding to the layered and Maxwell Garnett

composite geometries.

2.4.1 Lorentz local field in a resonant medium

In present work we consider composite materials consisting of two homogeneous con-

stituents. When studying local-field effects in composite systems one should account

for the local field in the constituents, together with the local field modifications im-

posed by the geometry of the composite. Within this study, we limit ourselves to the

case in which the homogeneous constituents obey the Lorentz model of the local field.

One can easily extend our formalism to the case of Onsager-type constituents.

We first consider a homogeneous medium with pure resonant molecules or atoms

(we call them emitters) illuminated by an optical field of frequency ω. We assume

that the field is tuned close to a resonance of the emitters. In that case we can


treat the medium roughly as a collection of two-level atoms, and can apply a well-

known formalism, based on the Maxwell–Bloch equations [41], to describe the optical

properties of such a medium. This formalism is described in Section 1.4. We insert

the Lorentz local field (2.1) into the Maxwell–Bloch Equations (1.19) for the slowly

varying amplitude σ(1) of the coherence σ(1)(t) = σ(1) exp (−iωt)+c.c. and population

inversion w to obtain:

˙σ(1) =

(

i∆ + i∆Lw − 1

T2

)

σ(1) − 1

2iκwE, (2.38a)

w = −w − weq

T1

+ i(

κE(σ(1))∗ − κ∗E∗σ(1))

. (2.38b)

Here ∆ = ω−ω0 is the detuning of the optical frequency ω from the atomic resonance

frequency ω0, T1 and T2 are the population and coherence relaxation times, respec-

tively, weq is the equilibrium value of the population inversion, and κ = 2µ/~, where

µ is the dipole moment of the atomic transition. Here and below, we use a tilde to

denote the time-dependent quantities, and similar variables without a tilde to denote

their slowly-varying amplitudes. Within this study we are only concerned with the

linear optical response of the laser gain media. In order to emphasize it, we added

the superscript “(1)” to the coherence σ(1). The inversion-dependent frequency shift

∆Lw involves a shift of the resonance of the atomic transition. Here ∆L is known as


Lorentz red shift [45] and is given by

∆L = −4π

3

N |µ|2~

. (2.39)

The slowly-varying amplitude P of the macroscopic polarization can be expressed

in terms of the linear susceptibility χ(1) as P = χ(1)E. We find from Eq. (1.20) that

χ(1) =Nµ∗σ(1)

E. (2.40)

Substituting the steady-state solution of Eq. (2.38a) for the coherence σ(1), given by

Eq. (1.24b) in Section 1.4, yields

χ(1) =N |µ|2

~

weq

∆ + ∆Lweq + i/T2

. (2.41)

Here weq is the equilibrium value of the population inversion (e.g., in the case of an

uninverted system, weq = −1). Now we can find small-signal gain (g0) and absorption

(α0) coefficients of the medium from Eq. (2.17):

g0 = −α0 = −4πω

c√

ǫIm χ(1). (2.42)

The dielectric permittivity ǫ entering Eq. (2.42) describes the dielectric properties of

the entire material structure away from its resonances.

The above analysis is valid for the case of “pure resonant emitters” (PRE) in


which all emitters are of the same sort. However, when dealing with laser gain media,

it is more typical to have a system with atoms or molecules of two or more sorts.

Therefore, it is useful to modify the above two-level model to consider the case in

which one has a background (host) material with the transition frequencies far from

that of the optical field, doped with some portion of atoms with a transition frequency

in resonance with the optical field. We refer to this case as “resonant emitters in a

background” (REB). For this kind of medium, we can split the total polarization

entering Eq. (2.1) for the Lorentz local field (we call it Ptot from now on) into a

contribution coming from the atoms of the background medium and a contribution

from the resonant atoms:

Ptot = Nbg αbg Eloc + Nres µ∗res σ(1)

res , (2.43)

where αbg is the polarizability of a background atom. Here and below the parameters

with the subscripts “bg” and “res” refer to the background and resonant atoms or

molecules of the medium. As we consider only media in which the Lorentz model of

local field is valid, both resonant and background types of molecules or atoms should

experience the same Lorentz local field.

One can relate the polarizability αbg to the dielectric constant of the background

material ǫbg using the Clausius–Mossotti (or Lorentz–Lorenz) relation [6, 37,40]

ǫbg − 1

ǫbg + 2=

4π

3Nbg αbg, (2.44)


also obtained in Section (1.3.1) (see Eq. (1.12)). Substituting Eq. (3.17) into Eq. (2.1)

for the local field and using Clausius–Mossotti relation (2.44) and Eq. (2.2) for the

local-field correction factor, we find

Eloc = Lbg

(

E +4π

3Pres

)

. (2.45)

The current expression for the local field reduces to Eq. (2.1) if one considers a vacuum

to be a background medium.

Next, we find the susceptibility of the medium from the relationship χ(1) = Ptot/E.

Substituting Eq. (3.17) and the steady-state solution (1.24b) for σ(1)res into the above

relationship for χ(1), we arrive at the result

χ(1) = χ(1)bg +

Nres |µres|2~

L2bg weq, res

∆ + ∆′L weq, res + i/T2, res

(2.46)

for the linear susceptibility of the REB medium. The frequency shift ∆′L is a modified

Lorentz red shift, given by

∆′L = −4π

3

Nres |µres|2~

Lbg = Lbg ∆L. (2.47)

This enhancement of the Lorentz red shift by Lbg due to the influence of the back-

ground dielectric medium had been previously noted in [115].


2.4.2 Linear Susceptibility of Layered Composite Materials

Layered composite materials [10, 27] are periodic structures consisting of alternating

layers of two or more homogeneous materials with different optical properties. We

presented an overview of layered composite materials in Section 1.2.1. Here we con-

sider a layered composite material comprised of two types of homogeneous media (we

call them a and b) with different optical properties (see Fig. 1.1(c)).

Let us assume that both components a and b of our layered composite material

respond linearly to the applied optical field. Assume that the field of frequency ω is

tuned close to one of the resonances of the component a, but does not coincide with

any of the resonances of the component b. We further follow the recipe given in [27]

for deriving the local field in a layer a.

We choose an axis Z to be perpendicular to the layers and focus on describing

the Z-component of the electric field and polarization. We define E and P to be the

average macroscopic field and polarization in the composite material, respectively, ea

and eb to denote mesoscopic fields in layers a and b, respectively, Ea, loc as a local field

in layer a, and pa and pb as mesoscopic polarizations in layers a and b, respectively.

For simplicity of notation, we do not write the indices z, indicating Z-components,

next to the fields and polarizations. Following the results obtained in [27] we can

write

ej = E + 4πP − 4πpj, (2.48)

where j = a, b for the mesoscopic fields in layers a and b. The macroscopic polarization


is related to the mesoscopic polarizations in layers a and b as

P = fa pa + fb pb (2.49)

with the volume fractions obeying the relationship fa +fb = 1. As the applied optical

field’s frequency does not coincide with any of the resonances of the medium b, we

can simply write

pb = χ(1)b eb (2.50)

for the mesoscopic field and polarization in a layer b, with χ(1)b denoting the linear

susceptibility in the layer. Using Eq. (2.50) and the relationship ǫb = 1 + 4πχ(1)b

for the dielectric constant and susceptibility in layer b, we obtain from Eq. (2.48)

the expressions for the mesoscopic fields in terms of the average field E and the

mesoscopic polarization pa in the form

ea =ǫb

1 + 4πfa χ(1)b

E − 4πfb

1 + 4πfa χ(1)b

pa (2.51a)

and

eb =1

1 + 4πfa χ(1)b

E +4πfa

1 + 4πfa χ(1)b

pa. (2.51b)

Next we want to express the local field Ea, loc in a layer a in terms of the average

field E and the mesoscopic polarization pa. Here we distinguish two cases: the case of

PRE, in which all the atoms or molecules of the material a are of the same sort, and


the case of REB, in which only a part of the atoms or molecules of the homogeneous

medium a are in resonance with the optical field.

Case of pure resonant emitters

We first treat the simpler case of PRE in the constituent a. In this case, the local

field in the medium a takes the form

Ea, loc = ea +4π

3pa. (2.52)

Substituting Eq. (2.51a) into Eq. (2.52), we find the expression for the local field in

terms of the average field and the mesoscopic polarization in a layer a,

Ea, loc =ǫb

1 + 4πfa χ(1)b

E +4π

3

fa ǫb − 2fb

1 + 4πfa χ(1)b

pa. (2.53)

The effective susceptibility of the layered composite material can be found from

χ(1)eff = P/E. Substituting pb in the form of Eq. (2.50) and pa in the form of Eq. (1.20)

into Eq. (2.49) for the macroscopic polarization and using Eq. (2.51b) for eb and

the steady-state solution (1.24b) for the coherence σ(1)a , we finally obtain the expres-

sion for the effective susceptibility of a layered composite material with PRE in the

constituent a:

χ(1)eff =

fa ǫ2b

(1 + 4πfa χ(1)b )2

Na |µa|2~

weq, a

∆ + ∆l weq, a + i/T2, a

+fb χ

(1)b

1 + 4πfa χ(1)b

. (2.54)


The local-field-induced frequency shift ∆l of the resonance feature has the form

∆l = −4π

3

|µa|2Na

~

fa ǫb − 2fb

1 + 4πfa χ(1)b

. (2.55)

The subscript “a” next to the atomic parameters indicates that they refer to the

resonant constituent a. One can easily verify that in the limiting case of fa → 1 and

fb → 0 Eq. (2.54) reduces to Eq. (2.41) for the susceptibility of a pure homogeneous

medium, and Eq. (2.55) takes the form of Eq. (2.39) for the Lorentz red shift. In the

opposite limiting case of fa → 0 and fb → 1, the effective susceptibility reduces to

χ(1)b .

Case of resonant emitters in a background

Now we consider the case of REB in layers a. In that case, the local field Ea, loc in

the layer a can be expressed as

Ea, loc = ea +4π

3(pa, bg + pa, res). (2.56)

The background and resonant contributions to the total mesoscopic polarization of

the material a are given by

pa, bg = Na, bg αa, bg Ea, loc (2.57a)


and

pa, res = Na, res µ∗a, res σ(1)

a, res. (2.57b)

Substituting Eqs. (2.57a) and (2.51a) into Eq. (2.56) yields the expression for the local

field in terms of the average field and the resonant contribution to the mesoscopic

polarization,

Ea, loc =La, bg ǫb

1 + 4π(fa χ(1)b + fb χ

(1)a, bg)

E +4π

3

La, bg(1 + 4πfa χ(1)b − 3fb)

1 + 4π(fa χ(1)b + fb χ

(1)a, bg)

pa, res. (2.58)

We find the effective susceptibility from χ(1)eff = P/E, using Eqs. (2.49), (2.50),

(2.51b), (2.57), (2.58), and the steady-state solution (1.24b) for coherence σ(1)a, res:

χ(1)eff = fa

ǫb

1 + 4πfa χ(1)b

ǫb

1 + 4π(fa χ(1)b + fb χ

(1)a, bg)

×[

χ(1)a, bg + La, bg

(

4π

3

χ(1)a, bg(1 + 4πfa χ

(1)b − 3fb)

1 + 4π(fa χ(1)b + fb χ

(1)a, bg)

+ 1

)

(2.59)

× Na, res |µa, res|2~

weq, res

∆ + ∆′l weq, res + i/T2, res

]

+fb χ

(1)b

1 + 4πfa χ(1)b

.

The frequency shift ∆′l of the resonant feature is given by

∆′l = −4π

3

|µa, res|2 Na, res

~

La, bg (1 + 4πfa χ(1)b − 3fb)

1 + 4π(fa χ(1)b + fb χ

(1)a, bg)

. (2.60)

In the limit fa → 1 and fb → 0 Eq. (2.59) reduces to Eq. (2.46), and the frequency

shift (2.60) reduces to the modified Lorentz red shift (2.47). In the opposite limiting


case of fa → 0 and fb → 1 we obtain χ(1)eff = χ

(1)b .

2.4.3 Linear Susceptibility of Maxwell Garnett Composite

Materials

The Maxwell Garnett type of composite geometry is a collection of small particles

(the inclusions) distributed in a host medium (see Section 1.2.1, Fig. 1.1(a)). The

effective dielectric constant of a Maxwell Garnett composite material is described by

Eq. (1.1).

Here we derive the local-field-corrected total susceptibility function for a Maxwell

Garnett composite material with homogeneous host and inclusion materials. Because

of the geometry of Maxwell Garnett composite materials, the local field is uniform

in the inclusion medium and non-uniform in the host [25]. We limit ourselves to

treating the case of resonant species in inclusions, as treating a more complicated

case of resonance in a host is beyond the scope of the present work. Following our

recipe, one can numerically solve the problem to deduce the total susceptibility and

associated frequency shift of the resonant feature for the latter case.

Assuming that both the host and inclusion materials respond linearly to the ap-

plied optical field and following the prescriptions given in the Ref. [25], we derive

Eq. (A.8) for the mesoscopic electric field ei in an inclusion in the Appendix A. Here

we drop the vector notation, as the inclusion medium is assumed to be isotropic and


uniform:

ei =3ǫh

3ǫh − 4πfh χ(1)h

[

E − 4π

3ǫh

fh pi

]

. (2.61)

In the above equation χ(1)h and fh are the linear susceptibility and volume fraction of

the host, respectively, and pi is the mesoscopic polarization in an inclusion. As in the

previous subsection, we consider the cases of PRE and REB in inclusions.

Case of pure resonant emitters

We obtain the expression for the local field Ei, loc acting on the emitters in an inclusion

from Eq. (2.1):

Ei, loc = ei +4π

3pi. (2.62)

Substituting Eq. (2.61) yields

Ei, loc =3ǫh


E +4π

3

ǫh(2 + fi) − 2fh


pi. (2.63)

Substituting the local field (2.63) into the Maxwell–Bloch equations (1.19), we find

their steady-state solutions.

The macroscopic polarization P is an average of the mesoscopic polarization p(r):

P =

∫

∆(r − r′)p(r′) dr′. (2.64)

The weighting function ∆(r) is defined in Ref. [25] and in the Appendix A. Introduc-


ing the functions

p(r) =

pi(r) if r ∈ inclusion,

ph(r) if r ∈ host

(2.65)

We can represent the macroscopic polarization as an average of the mesoscopic po-

larizations pi and ph in the inclusions and host:

P =

∫

∆(r − r′)pi(r′) dr′ +

∫

∆(r − r′)ph(r′) dr′. (2.66)

We can represent the macroscopic electric field E as an average of the mesoscopic

electric fields in the inclusion and host in a similar way. Since the mesoscopic polar-

ization and electric field in an inclusion are uniform, the first terms on the right-hand

side of Eq. (2.66) is equal to fi pi. Similarly, one can obtain fi ei for the electric field.

Using this result, we find the expression for the macroscopic polarization:

P = fi pi(r) + χ(1)h E − fi χ

(1)h ei(r). (2.67)

Substituting Eq. (2.61) for ei, Eq. (1.20) for the resonant pi, and the steady-state

solution for σ(1)i to Eq. (2.67) for the macroscopic polarization, and using the rela-

tionship χ(1)eff = P/E, we finally obtain the expression for the effective susceptibility


of a Maxwell Garnett composite material with a resonance in the inclusions:

χ(1)eff = fi

(

3ǫh


)2Ni |µi|2

~

weq, i

∆ + ∆MGweq, i + i/T2

+ fhχ

(1)h (2ǫh + 1)

3ǫh − 4πfhχ(1)h

. (2.68)

The Maxwell Garnett frequency shift ∆MG of the resonance feature has the form

∆MG = −4π

3

Ni |µi|2~

ǫh(2 + fi) − 2fh


. (2.69)

The subscript “i” next to the atomic parameters indicates that they refer to the

inclusion material. It is easy to verify that Eq. (2.68) reduces to Eq. (2.41), and the

frequency shift (2.69) reduces to the Lorentz red shift (2.39) in the limit fi → 1 and

fh → 0. We obtain χ(1)eff = χ

(1)h in the opposite limiting case.

Case of resonant emitters in a background

Here we consider the case of REB in inclusions. The local field Ei, loc acting on an

emitter in an inclusion is given by Eq. (2.63) in terms of the macroscopic field E and

the mesoscopic polarization pi in an inclusion, but we need to express it in terms of

the field E and the resonant part pi, res of the polarization pi, which is now

pi = pi, bg + pi, res. (2.70)


The background non-resonant and resonant contributions to the mesoscopic polariza-

tion of an inclusion are given by

pi, bg = Ni, bg αi, bg Ei, loc (2.71a)

and

pi, res = Ni, res µ∗i, res σ

(1)i, res. (2.71b)

Substituting Eqs. (2.70) and (2.71) into Eq. (2.63) yields the expression

Ei, loc =3ǫh Li, bg

3ǫh + 4πfh (χ(1)i, bg − χ

(1)h )

E +4π

3Li, bg

ǫh (2 + fi) − 2fh


(1)h )

pi, res (2.72)

for the local field in terms of the macroscopic field and resonant contribution to the

mesoscopic polarization. Substituting the local field in the form of Eq. (2.72) into

the Maxwell–Bloch equations (1.19), we find the steady-state solution for coherence

σ(1)i, res.

One can find the effective susceptibility from χ(1)eff = P/E by substituting P from

Eq. (2.67) with ei given by Eq. (2.61) and pi given by Eqs. (2.70) and (2.71), and the


steady-state solution for σ(1)i, res:

χ(1)eff = fi

3ǫh


3ǫh


(1)h )

×[

χ(1)i, bg + Li, bg

(

4π

3

χ(1)i, bg [ǫh(2 + fi) − 2fh]


(1)h )

+ 1

)

(2.73)

× Ni, res |µi, res|2~

weq, res

∆ + ∆′MGweq, res + i/T2, res

]

+ fh χ(1)h

2ǫh + 1


.

The frequency shift ∆′MG is given by

∆′MG = −4π

3

Ni, res |µi, res|2~

Li, bg [ǫh (2 + fi) − 2fh]


(1)h )

. (2.74)

In the limit fi → 1 and fh → 0 Eq. (2.73) reduces to Eq. (2.46), and the frequency

shift (2.74) reduces to the modified Lorentz red shift (2.47). In the opposite limit,

χ(1)eff = χ

(1)b .

2.4.4 Analysis

In this subsection we analyze the results for the effective linear susceptibility obtained

in the previous subsections for the layered and Maxwell Garnett composite materials.

Layered Geometry

The local-field-corrected effective linear susceptibility for a layered composite material

with the resonance in the layers a is given by Eq. (2.54) for the case in which all the


atoms or molecules of a are of the same sort (PRE case), and by Eq. (2.59) for the

case in which the component a consists of particles of different sorts (REB case).

One can find the small-signal gain coefficient of the layered composite material by

substituting an expression for the effective susceptibility into Eq. (2.42).

Here we analyze the behavior of the small-signal gain as a function of various pa-

rameters, choosing a Rhodamine 6G-doped PMMA laser gain medium as the resonant

species a in our layered composite material. The parameters of the gain medium that

we used for our analysis are the emission peak wavelength λ0 = 590 nm, the trans-

verse relaxation time T2 = 100 fs, the transition cross section σtr = 2×10−16 cm2, the

Rhodamine molecular concentration N = 1.8 × 1018 cm−3, and the refractive index

of PMMA, nbg = 1.4953. We take the component b to be an unknown material and

vary its refractive index to see how it affects the optical response of Rhodamine-doped

PMMA.

Rhodamine-doped PMMA is an example of the REB case, and, therefore, one

should use Eq. (2.59) in order to describe the effective susceptibility of the system

with Rhodamine-doped PMMA resonant layers. We compare the model given by

Eq. (2.59) to the case of PRE, described by Eq. (2.54), in order to test whether one

can use the latter approximation in case of REB. Using the PRE model in this physical

case implies that we neglect the presence of PMMA, assuming that the Rhodamine

molecules are sitting in a vacuum.

We also find it informative to compare the results given by the more precise REB


model (2.59) to that of the simplified model that we proposed in Section 2.2 (see

Eq. (2.28b)). The approximation underlying the simplified model is that a compos-

ite laser gain medium is represented as a quasi-homogeneous medium characterized

by an effective refractive index, and the local-field effects are accounted for by the

Lorentz model for homogeneous media. For the purpose of comparison, we adapt this

simplified model to our REB case in the following way. We take the expression (2.46)

for the linear susceptibility of a medium with REB, substituting χ(1)eff , deduced from

Eq. (1.4) for ǫeff of a layered composite material, in place of χ(1)bg , and Leff = (ǫeff +2)/3

in place of Lbg. We also multiply the atomic densities appearing in the expression by

the volume fraction fa to take into account the fact that, changing the volume fraction

of the resonant component, we change the density of the resonant molecules. Then

we substitute the resulting expression for χ(1) to Eq. (2.28b) to find the small-signal

gain coefficient, corresponding to the simplified model.

Setting the equilibrium value of the population inversion weq = 1, which corre-

sponds to a fully inverted amplifying system, and the detuning of the optical field

with respect to the resonance ∆ = 0, we plot the small-signal gain calculated using

the effective susceptibility given by Eq. (2.59) as a function of the refractive index of

the non-resonant component nb in Fig. 2.7(a), and of fa in Fig. 2.8. In addition, we

plot the gain coefficients derived from Eq. (2.54) and from the simplified model from

Section 2.2 as functions of nb in Figs. 2.7(b) and 2.7(c), respectively. The compar-

ison between parts (a) and (b) of Fig. 2.7 indicates that the results for the cases of


0

100

200

300

400

500

1 1.5 2 2.5 3 3.5 4

g 0 (

cm-1

)

nb

(c)0

100

200

300

400

500

g 0 (

cm-1

)

(b)0

100

200

300

400

500

600

g 0 (

cm-1

)

(a)

fa0.00.20.40.60.81.0

Figure 2.7: Small-signal gain of a layered composite material as a function of the

refractive index of the non-resonant component for different values of the volume

fraction of the resonant component: (a) REB case [derived from Eq. (2.59)], (b)

PRE case [from Eq. (2.54)], (c) the simplified model.

PRE and REB differ significantly, which means that the approximation based on the

assumption that all the atoms or molecules of the resonant species are of the same

sort does not work in the REB case. The comparison of Figs. 2.7(a) and 2.7(c)


0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1

g 0 (

cm-1

)

fa

nb1.01.51.82.02.53.04.0

Figure 2.8: Small-signal gain of a layered composite material as a function of

the volume fraction of the resonant component for different values of the refractive

index of the non-resonant component. The gain is derived from Eq. (2.59) for the

REB case.

shows moreover that the simplified model is insufficiently precise. Therefore, as we

stated earlier in Section 2.2, it is only good enough for a proof-of-principle study,

and separate, more precise models for each composite geometry are required for an

accurate description of the optical response.

It can be seen from Figs. 2.7 and 2.8 that the gain coefficient tends to grow with

the increase of the refractive index of the non-resonant component. The reason for this

behavior is that for the light polarized perpendicular to the layers, the electric field

tends to localize in the regions of a dielectric with lower refractive index [27]. There-

fore, the higher refractive index of the non-resonant layers is, the more the electric

field is displaced into the resonant layers, which causes a stronger gain. The behavior

of the gain coefficient as a function of fa is more complex. It grows monotonously

with the increase of fa for small refractive indices. In the case nb = nbg the growth is


linear. For high values of the refractive index the small-signal gain displays a rapid

growth with the increase of the volume fraction until it reaches a maximum value,

corresponding to an optimal value of fa, after which it starts to decrease with further

increase of fa. This behavior can be understood as follows. The initial growth of g0

with fa is due to the fact that the number of the resonant molecules in the medium

increases. On the other hand, increasing fa, we make our layers with the resonant

molecules thicker, and the local field, highly concentrated in these layers because of

the high value of nb, spreads over the layers, and each individual molecule “feels” a

smaller value of the local field. This causes the gain to decrease with the increase of

fa beyond an optimal value. Thus, in order to achieve maximum gain or absorption

in a layered composite material, one needs to use a non-resonant component with a

high refractive index, while keeping the volume fraction of the resonant component

low.

Setting fa = 0.5 and nb = 1.8, we plot in Fig. 2.9 the small-signal gain for REB

in the component a as a function of the frequency detuning ∆ of the optical field

with respect to the molecular resonance for different values of weq. Variation of

the population inversion from weq = −1, corresponding to an uninverted system, to

weq = 1, describing a fully inverted system, clearly shows how the composite medium

changes from an absorber to an amplifier. In reality, it is not possible to achieve a

full inversion, so, most physical cases correspond to weq < 0.5.


-400

-300

-200

-100

0

100

200

300

400

-4 -3 -2 -1 0 1 2 3 4

g 0 (

cm-1

)

∆ (sec-1 × 1013)

weq = 1.0

0.50.250.0

-0.25-0.5

-1.0

Figure 2.9: Small-signal gain of a layered composite material as a function of

the detuning of the optical field with respect to the molecular resonant frequency.

Marked on the graphs are the values of equilibrium population inversion.

Maxwell Garnett Geometry

The linear effective susceptibility for the Maxwell Garnett composite geometry for

the case of PRE and REB in inclusions is given by Eqs. (2.68) and (2.73), respec-

tively. Substituting these expressions into Eq. (2.17), we obtain the small-signal gain

coefficients for the cases of PRE and REB.

We take for our analysis a Maxwell Garnett composite material with Nd:YAG

nanoparticles as inclusions. We use the emission wavelength λ0 = 1.064 µm, the

transverse relaxation time T2 = 3 ms, the transition cross section σtr = 4.6×10−19cm2,

the neodymium atomic concentration N = 1.37 × 1020 cm−3 (this value corresponds

to 1 at. % of Nd in YAG), and the YAG refractive index nbg = 1.82. We take the

host to be an unknown medium, and vary its refractive index nh to see how it affects

the optical response of Nd:YAG nanoparticles.


Nd:YAG corresponds to the case of REB, which means that the effective suscep-

tibility of the Maxwell Garnett composite material with Nd:YAG inclusions obeys

Eq. (2.73). As in the previous section, we compare the small-signal gain coefficients,

corresponding to the REB model, given by Eq. (2.73), to the PRE model, given by

Eq.(2.68), and to the simplified model from Section 2.2, adapted to our case, in order

to test the applicability of the latter two approximations. The simplified model is de-

rived from Eq. (2.46) with χ(1)eff and Leff , deduced from Eq. (1.1) for ǫeff of the Maxwell

Garnett composite material, in place of χ(1)bg and Lbg. In addition, we multiply the

atomic density in the resulting equation by the inclusion volume fraction fi in order

to account for its change with the change of fi. Then we substitute the resulting

expression for χ(1) into Eq. (2.17) to deduce the gain coefficient.

Setting the equilibrium value of the population inversion weq = 1 and the detuning

∆ = 0, we plot the small-signal gain coefficients derived from the REB [Eq. (2.73)]

and PRE [Eq. (2.68)] models, and from the simplified model, as functions of the

refractive index of the host nh in Figs. 2.10(a), 2.10(b), and 2.10(c), respectively.

The dependence of g0, derived from Eq. (2.73), on the inclusion volume fraction fi

is depicted in Fig. 2.11. A comparison of the part a with the parts b and c of

Fig. 2.10 suggests that, as in the case of the layered composite geometry, the PRE

approximation and simplified model do not agree with the more precise description

of the Maxwell Garnett composite material with REB in inclusions.

One can see from Figs. 2.10 and 2.11 that the small-signal gain of the Maxwell


0

20

40

60

80

100

1 1.5 2 2.5 3 3.5 4

g 0 (

cm-1

)

nh

fi

(c)

0.00.20.40.60.81.0

0

20

40

60

80

100

g 0 (

cm-1

)

(b)0

20

40

60

80

100

120

g 0 (

cm-1

)

(a)

Figure 2.10: Small-signal gain of a Maxwell Garnett composite material as a

function of the refractive index of the non-resonant host for different values of the

inclusion volume fraction: (a) REB case [from Eq. (2.73)], (b) PRE case [from

Eq. (2.68)], (c) the simplified model from.

Garnett composite geometry exhibits a monotonic growth with the increase of fi. It

increases to some maximum value with the increase of the host refractive index, and

then decreases with further growth of nh. The monotonic growth of g0 with fi is due


0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1

g 0 (

cm-1

)

fi

nh1.01.51.82.02.53.04.0


function of the inclusion volume fraction for different values of the host refractive

index. g0 is derived from Eq. (2.73) for REB case.

to the fact that, unlike in layered composite materials, the increase of the inclusion

volume fraction in the Maxwell Garnett composite material is not accompanied by

the decrease in the local field in an inclusion. The reason to the complex behavior of

g0 as a function of nh is as follows. It is seen from Eq (2.17) that g0 ∝ [√

ǫeff ]−1Im χ(1)eff .

Due to the electric field localization in the component with the lower refractive index,

Im χ(1)eff monotonically grows with the increase of nh. However, the term [

√ǫeff ]−1 ,

where ǫeff is given by Eq. (1.1), decreases with the increase of nh, and at some value

of the host refractive index its decrease overcompensates the growth of Im χ(1)eff . As a

result, g0 starts to decrease with further increase of nh.

It is important to keep in mind that the Maxwell Garnett model works well only for

low volume fractions of the inclusions (fi . 0.5). It does not account for a percolation

phenomenon that occurs when fi is higher than a certain value.


In Fig. 2.12 we plot the small-signal gain coefficient as a function of the detuning

∆ for different values of the equilibrium population inversion weq with the fixed values

fi = 0.01 and nh = 1.3. As in the case of the layered composite geometry, one can

-1

-0.5

0

0.5

1

-1000 -500 0 500 1000

g 0 (

cm-1

)

∆ (sec-1)

-1.0-0.5

-0.250.0

0.250.5

weq = 1.0


function of the detuning of the optical field with respect to the molecular resonant

frequency. Marked on the graphs are the values of equilibrium population inversion.

see the change from absorption to amplification as weq changes from -1 to 1.

The work reported in Section 2.4 is going to be submitted to the Journal of Physics

A [116].

2.5 Conclusions

We have proposed a method for controlling and tailoring the basic laser properties

of laser gain media, such as the radiative lifetime of the upper laser level, the small-

signal gain coefficient, and the saturation intensity. The idea behind the method is


designing composite materials by mixing two or more materials on a nanoscale to

obtain new laser gain media with the laser properties enhanced compared to those

of the constituents. The enhancement can be achieved by implementing local-field

effects that can significantly modify optical properties of materials. The composite-

material approach has been used for enhancing the nonlinear properties of dielectric

[25–29, 31, 32] and metal-dielectric [30, 117] composites. However, we are unaware of

any previous systematic study of the modification of the laser properties of composite

materials by the local-field effects. Our idea of composite laser has been published in

the the Journal of the Optical Society of America [105].

Together with the simple proof-of-principle study, we also developed more sophis-

ticated theoretical models describing the linear effective susceptibility of layered and

Maxwell Garnett composite materials with resonant components of two types. The

first type is a simple case of a resonant medium in which all the molecules or atoms are

of the same sort. The second type of resonant component corresponds to a situation

in which only a fraction of the molecules or atoms of the medium are in resonance

with the optical field. The latter model is more realistic, as most of the laser gain

media consist of species of different sorts. Along the way, we derived the expressions

for the frequency shifts of the resonant features with respect to the actual resonances,

which are the analogs of the famous Lorentz red shift in a homogeneous medium.

These frequency shifts for the layered and Maxwell Garnett composite geometries

with pure resonant emitters and resonant emitters in a background differ from the


Lorentz red shift and from each other.

We analyzed our theories of Layered and Maxwell Garnett composites for the

following two physical cases. The first case is a layered composite material with the

resonant component consisting of Rhodamine-doped PMMA. The second case is a

Maxwell Garnett composite material with Nd:YAG nanoparticles as resonant inclu-

sions. The layered geometry seems to be more promising for tailoring the optical

properties of the composite laser gain media: under certain conditions the gain co-

efficient of the layered composite geometry can exceed the gain coefficients of its

constituents. The theoretical models for layered and Maxwell Garnett composite

geometries are to be submitted to the Journal of Physics A [116].

We have also experimentally investigated the influence of local-field effects on

the radiative lifetimes of liquid suspensions of Nd:YAG nanoparticles. To determine

which model (the real-cavity, the virtual-cavity model, or no-local-field-effects model)

gives the best description of our experimental data, we compared the data with the

predictions of these models. We find that we can rule out the virtual cavity model

as it is in obvious disagreement with our data for any of the values of the quantum

yield that we considered. We also find that the real cavity model can be used to

provide a good fit to our data for the range of possible quantum yields. However, we

cannot rule out the no-local-fields model on experimental grounds. If we assume that

quantum yield is unity, the no-local-fields model is also in good agreement with our

data. However, there is no theoretical reason to believe that local field effects would


not occur in these materials, and for this reason we feel that the real-cavity model

best describes our experimental results. This work has been published in the Journal

of the Optical Society of America [24].

Chapter 3

Microscopic Cascading in

Fifth-Order Nonlinear

Susceptibility Induced by

Local-Field Effects

3.1 Introduction

The development of lithographic materials that can allow one to overcome the Rayleigh

limit is crucial for improving the spatial resolution in optical lithography. Many ap-

proaches have been undertaken in this direction. It has been realized [118] and exper-

imentally demonstrated [119] that N -photon absorbing materials allow one to achieve

84

CHAPTER 3. MICROSCOPIC CASCADING 85

N -fold resolution enhancement in interference lithography. Boto, et al. [118] proposed

to expose an N -photon lithographic material to an interference pattern produced by

a quantum-mechanically entangled source. This procedure would allow one to record

a fringe pattern with unit visibility and a period N times smaller than the Rayleigh

λ/2 limit. However, realization of the idea in the laboratory remains a challenge for

experimentalists working in the field of quantum imaging.

Here we propose a novel principle for designing N -photon absorbing lithographic

materials and materials with tailored high-order nonlinear response based on micro-

scopic cascaded nonlinear effects.

Cascaded nonlinearities have proven to be very useful in the field of nonlinear

optics because they allow a sequence of low-order responses to mimic a high-order

response. Because lower-order nonlinearities are typically much stronger than higher-

order nonlinearities, cascaded lower-order processes can be much more efficient than

higher-order processes. It is useful to distinguish macroscopic cascading from mi-

croscopic cascading. Macroscopic cascading occurs as a result of propagation ef-

fects [62, 65, 67]. Somewhat more subtle is the origin of microscopic cascading, in

which higher-order effects are induced in the response at the molecular level [60,68].

In this chapter we show both theoretically and experimentally that there are

microscopic cascaded effects in high-order nonlinearities, induced by local-field effects,

that can be very significant and useful for various applications. In Section 3.2 we

present a theoretical analysis of the nonlinear response of a medium, treated up to the


fifth order of nonlinearity, with local-field effects taken into account. We first address

this problem by solving the Lorentz–Maxwell–Bloch equations (see Section 1.4). We

expand the total susceptibility as a Taylor series in the electric field. We find that

the resulting expression for the fifth-order nonlinear susceptibility is in disagreement

with the straight-forward generalization of Bloembergen’s result [55] to fifth-order

response. We resolve this apparent contradiction by solving the problem using a more

careful implementation of Bloembergen’s approach [68]. The detailed calculation

shows that there is no disagreement between the results of the local-field-corrected

Maxwell–Bloch equations and that careful implementation. Moreover, the results

show that there is a cascaded contribution coming from the third-order microscopic

hyperpolarizability, together with the naıvely expected fifth order nonlinear term.

This cascaded contribution is a consequence of local-field effects.

In Section 3.3 we analyze the relative values of the contributions from the fifth- and

third-order microscopic hyperpolarizabilities to χ(5), and we find that under certain

conditions the cascaded third-order contribution can be as large as the direct fifth-

order contribution.

In Section 3.4 we describe an experiment that allows us to separate the influence

of microscopic cascading from the more-well-known macroscopic cascading, and we

find conditions under which microscopic cascading is the dominant effect.


3.2 Theoretical Prediction of Microscopic Cascad-

ing

3.2.1 Maxwell–Bloch equations approach

A collection of two-level atoms with ground and excited states denoted respectively

by a and b, interacting with an optical field closely tuned to an atomic resonance

of the system, can be described by the Maxwell–Bloch equations (1.19). Accounting

for local-field effects using the Lorentz local field (1.6) leads to a modified form of

Eqs. (1.19), given by (1.22). Here we recall the steady-state solutions to Eqs. (1.22)

for the case weq = −1:

w = − 1

1 +|E|2/|E0

s |21 + T 2

2 (∆ + ∆Lw)2

; (3.1a)

σ =µ

~

wE

∆ + ∆Lw + i/T2

. (3.1b)

Here E0s is the saturation field strength, defined by Eq. (1.25), and ∆L is the Lorentz

red shift, given by Eq. (1.23).

As a consequence of the presence of the local-field-induced inversion-dependent

frequency shift ∆Lw in Eq. (3.1a), the steady-state solution for the population in-

version w becomes a cubic equation. In a certain range of parameters it has three

real roots with absolute values not exceeding unity. The existence of three different


physically meaningful solutions for the population inversion is associated with the

phenomenon of local-field-induced optical bistability [41, 43, 44], first discussed by

Hopf, Bowden, and Louisell [41]. In Appendix B we identify the parameter space in

which Eq. (3.1a) has multiple physical solutions, and show that it cannot be reached

for the example system that we consider in Section 3.3, the excitation of a collection

of sodium atoms at frequencies close to the 3s → 3p resonance.

For such systems, the nonlinear response can be studied through an approximate

solution of Eq. (3.1a), obtained through a power-series expansion with respect to

the electric field parameter x = |E|2/|E0s |2. Assuming x to be a small quantity, we

perform a Taylor series expansion of w in terms of x, retaining only terms up to the

second power, as we are interested only in treating saturation effects up to the fifth

order in E. The resultant solution for w takes the form

w = −1 +1

1 + T 22 (∆ − ∆L)2

|E|2|E0

s |2− 1 + T 2

2 (∆2 − ∆2L)

[1 + T 22 (∆ − ∆L)2]

3

|E|4|E0

s |4. (3.2)

The total susceptibility χ, including linear and nonlinear interactions, is obtained

in Section 1.4 and is given by Eq. (1.27), which we recall here:

χ =N |µ|2T2

~

w

T2(∆ + ∆Lw) + i=

N |µ|2~

(−w)

(ωba + ∆L(−w) − ω) − i/T2

. (3.3)

Eqs. (3.2) and (3.3) illustrate the physical effect of the nonlinearity: it is through the

modification of the inversion parameter w from its equilibrium value of −1, and that


modification is twofold. First, the overall amplitude of the response is modified by the

fact that −w differs from unity, and second, the resonant frequency is modified from

ωba + ∆L (the Lorentz-shifted low-intensity resonance frequency) to ωba + ∆L(−w).

It is convenient to represent the total susceptibility as a power series expansion

with respect to the electric field:

χ = χ(1) + 3χ(3)|E|2 + 10χ(5)|E|4 + . . . . (3.4)

Then, substituting the expansion of the population inversion (3.2) into Eq. (3.3)

and making use of the representation (3.4) of the total susceptibility, we find the

expressions for the linear and the nonlinear susceptibilities to be

χ(1) = −N |µ|2T2

~

T2(∆ − ∆L) − i

1 + T 22 (∆ − ∆L)2

, (3.5a)

χ(3)|E0s |2 =

N |µ|2T2

3~× (T2∆ + i) [T2(∆ − ∆L) − i]2

[1 + T 22 (∆ − ∆L)2]

3 , (3.5b)

and

χ(5)|E0s |4 = −N |µ|2T2

10~

(T2∆ + i) [1 − iT2∆L + T 22 (∆ − ∆L)(∆ + 2∆L)]

[1 + T 22 (∆ − ∆L)2]

3[T2(∆ − ∆L) + i]2

. (3.5c)


3.2.2 The Naıve Local-Field Correction

We next attempt to bring expressions (3.5) for the local-field-corrected susceptibilities

to the form of Bloembergen’s result. The straight-forward generalization [56, 57] of

the Bloembergen’s result to the case of the saturation effects [which are described by

an odd-order nonlinearity as χ(i) = χ(i)(ω = ω + ω − ω + . . .)] reads

χ(i) = Nγ(i)at |L|i−1L2, (3.6)

where γ(i)at is the ith-order microscopic hyperpolarizability (ineq1).

Using Eqs. (1.19) and (1.21) for an isolated atom in free space, we write

p(t) = p exp(−iωt) + c. c. (3.7)

with

p = µ∗σ (3.8)

for the atom’s dipole moment p. The Taylor series expansion for p with respect to

the electric field yields

p = γ(1)at E + 3γ

(3)at |E|2E + 10γ

(5)at |E|4E + . . . . (3.9)


Here

γ(1)at = −|µ|2T2

~

T2∆ − i

1 + T 22 ∆2

(3.10)

is the linear polarizability, and

γ(3)(at)|E0

s |2 =|µ|2T2

3~

T2∆ − i

(1 + T 22 ∆2)2

(3.11a)

and

γ(5)(at)|E0

s |4 = −|µ|2T2

10~

T2∆ − i

(1 + T 22 ∆2)3

(3.11b)

are the third-order and fifth-order microscopic hyperpolarizabilities, respectively.

The definition of the factor L in terms of the dielectric function ǫ(1) is given by

Eq. (1.15). The next step is to find an expression for the factor L in terms of the

detuning ∆ and the Lorentz red shift ∆L. Setting w = −1 in Eq. (3.3), we arrive at

χ(1) = −N |µ|2T2

~

1

T2(∆ − ∆L) + i, (3.12)

and so

ǫ(1) = 1 + 4πχ(1) = 1 − 4πN |µ|2T2

~

1

T2(∆ − ∆L) + i. (3.13)

Using (3.13) in (1.15), we obtain

L =T2∆ + i

T2(∆ − ∆L) + i. (3.14)


Making use of Eq. (3.10) for microscopic polarizability, Eqs. (3.11) for microscopic

hyperpolarizabilities, and Eq. (3.14) for factor L, we find that

Nγ(1)at L = χ(1) (3.15a)

of Eq. (3.5a),

Nγ(3)at |L|2L2 = χ(3) (3.15b)

of Eq. (3.5b), but

Nγ(5)at |L|4L2 6= χ(5) (3.15c)

of Eq. (3.5c). In fact,

Nγ(5)at |L|4L2 = χ(5) 1 + T 2

2 (∆ − ∆L)2

1 − iT2∆L + T 22 (∆ − ∆L)(∆ + 2∆L)

. (3.16)

Thus, the naıve local-field correction (3.6) in terms of L’s is in disagreement with the

correct result derived from the Maxwell–Bloch equations.

To find the origin of this disagreement, we address the problem of treating the sat-

uration up to the fifth order of nonlinearity, following the recipe suggested by Bloem-

bergen [55], rather than using the straight-forward generalization given by Eq. (3.6).

Our calculations are presented in the following subsection.


3.2.3 Bloembergen’s approach

The polarization P entering Eq. (2.1) is the total polarization, given by the sum of

the contributions proportional to first, third and fifth power of the local electric field

as

P = P (1) + P (3) + P (5) + . . . . (3.17)

Here

P (1) = Nγ(1)at Eloc, (3.18a)

P (3) = Nγ(3)at |Eloc|2Eloc, (3.18b)

and

P (5) = Nγ(5)at |Eloc|4Eloc. (3.18c)

Using Eq. (2.1) in (3.18a), we obtain

P (1) =ǫ(1) − 1

4π

[

E +4π

3P (3) +

4π

3P (5) + . . .

]

. (3.19)

The electric displacement vector D is defined as

D = E + 4πP = E + 4πP (1) + 4πP (3) + 4πP (5) + . . . . (3.20)


Substituting (3.19) into (3.20), we find that

D = ǫ(1)E + 4πPNLS, (3.21)

where

PNLS = L(P (3) + P (5) + . . .) (3.22)

is the nonlinear source polarization, introduced by Bloembergen [55].

Substituting expression (3.19) for the polarization P (1) into Eq. (3.17) for the total

polarization, we find that

P = χ(1)E + PNLS. (3.23)

Substituting Eq. (2.1) for the local field into Eqs. (3.18b) and (3.18c) for the polar-

izations P (3) and P (5) and dropping out the terms scaling with higher than the fifth

power of the electric field, we obtain

P (3) = 3Nγ(3)at |L|2L|E|2E

+[

24πN2(γ(3)at )2|L|4L2 + 12πN2|γ(3)

at |2|L|6]

|E|4E (3.24a)

and

P (5) = 10Nγ(5)at |L|4L|E|4E. (3.24b)

Note that P (3) contains terms proportional to the fifth power of the electric field.

Substituting (3.24) into (3.22), and (3.22) into (3.23), we find the total polarization


to be

P = χ(1)E + 3Nγ(3)at |L|2L2|E|2E

+[

24πN2(γ(3)at )2|L|4L3 + 12πN2|γ(3)

at |2|L|6L

+ 10Nγ(5)at |L|4L2

]

|E|4E + . . . . (3.25)

Alternatively, the total polarization can be represented as a Taylor series expansion

with respect to the average electric field as

P = χE = χ(1)E + 3χ(3)|E|2E + 10χ(5)|E|4E + . . . . (3.26)

Equating (3.25) and (3.26), we obtain

χ(1) = Nγ(1)at L, (3.27a)

χ(3) = Nγ(3)at |L|2L2, (3.27b)

and

χ(5) = Nγ(5)at |L|4L2 +

24π

10N2(γ

(3)at )2|L|4L3 +

12π

10N2|γ(3)

at |2|L|6L. (3.27c)

Using expression (3.14) for the local-field correction factor, obtained in Section 3.2.2,

one can show that Eqs. (3.27) for the local-field-corrected first, third, and fifth order


susceptibilities are equivalent to Eqs. (3.5), obtained using Maxwell-Bloch approach.

Thus, two different approaches – the Lorentz–Maxwell–Bloch equations and Bloem-

bergen’s approach – bring us to the same result for the local-field-corrected suscepti-

bilities. This is of course not surprising, since both approaches are just different ways

of implementing Bloembergen’s scheme.

The expressions for local-field-corrected χ(1) and χ(3) do not display any pecu-

liarity, while Eq. (3.27c) for χ(5) deserves special attention. The first term on the

right-hand side of the equation is due to a direct contribution from the fifth-order mi-

croscopic hyperpolarizability, while the two extra terms come from the contribution

of the third-order microscopic hyperpolarizability. These extra contributions are a

manifestation of local-field effects. We denote for convenience the direct contribution

to the fifth-order susceptibility as

χ(5)dir = Nγ

(5)at |L|4L2. (3.28)

Similarly, the sum of the second and third terms on the left-hand side of (36c) (the

microscopic cascaded contribution to χ(5)) can be denoted as

χ(5)micro =

12π

10N2[

2(γ(3)at )2|L|4L3 + |γ(3)

at |2|L|6L]

. (3.29)

Then the total local-field-corrected χ(5), which is the sum of the two contributions,


can be written as

χ(5) = χ(5)dir + χ

(5)micro. (3.30)

As we pointed out in preceding sections of this paper, the result that we obtained

for χ(5) does not agree with that predicted by a straight-forward generalization of the

Bloembergen’s result given by Eq. (3.6). It is evident from Eqs. (3.6) and (3.27c)

that the generalization (3.6) predicts the direct term only (the term proportional to

γ(5)at ) in the expression for the local-field-corrected χ(5), and does not account for the

cascaded contributions coming from the third-order microscopic hyperpolarizability.

We have shown in this section that the cascaded terms arise from substituting the

nonlinear local field into the expression (3.18b) for P (3). If we were limiting ourselves

to considering the third-order nonlinearity (i. e., the lowest-order nonlinearity in our

system), then we would need only to substitute the linear local field,

Eloc = E +4π

3P L,

into equation (3.18b) to deduce that

P (3) = 3Nγ(3)at |L|2L|E|2E,

instead of P (3) in the form of Eq. (3.24a). Thus, one clearly cannot simply use the

generalization (3.6) to treat nonlinearity of the order higher than the lowest order of

the nonlinearity present in the system of interest.


To develop insight into the relative contributions of the third- and fifth-order

hyperpolarizabilities to the local-field-corrected fifth-order susceptibility (3.27c), we

conduct a comparative analysis of the direct and cascaded terms. The analysis iden-

tifies the importance of the cascaded terms, and is presented in the following section.

3.3 Numerical Analysis

We perform our analysis based on a realistic example, taking the values of parameters

for the sodium 3s → 3p transition. The transition dipole moment is |µ| = 5.5 ×

10−18 esu and the population relaxation time is T1 = 16 ns. The value of the coherence

relaxation time T2 can be found according to [45]

γ2 =γnat

2+ γself .

Here γ2 = 1/T2 is the atomic linewidth, γnat = 1/T1 is the natural (radiative)

linewidth, and the collisional contribution γself is given by

γself =4πN |µ|2

~

√

2Jg + 1

2Je + 1,

where Jg and Je are the angular momentum quantum numbers of the ground and

excited states, respectively (for the sodium 3s → 3p transition, Jg = 0 and Je = 1).

In the theoretical analysis developed in Section 3.2 we have implicitly used the


rotating wave approximation (RWA) to describe the atomic response. Before pro-

ceeding here we confirm the validity of that approximation for our example of the

sodium 3s → 3p transition. We begin by comparing our RWA expression (3.12) to a

more precise expression in which the RWA approximation is not made,

χ(1)non−RWA =

N |µ|2~

[

1

(ωba − ω + ∆L) − i/T2

+1

(ωba + ω − ∆L) + i/T2

]

, (3.31)

where it is the second term in (3.31) that is missing in the RWA. Evaluating Re(χ(1)non−RWA)/Re(χ(1))

and Im(χ(1)non−RWA)/Im(χ(1)) over the range of atomic densities and frequency detun-

ings that we use in this section, we find that even at N = 1017 cm−3 the maximum

deviations of those ratios from unity are only a fraction of a percent. And even if the

ratios are raised to the third and fifth powers, as a rough sense of how the higher order

susceptibilities will be sensitive to expressions beyond the RWA, we find that their

maximum deviations from unity are at most a few percent. Thus we feel comfortable

in using the RWA in our identification of the range of validity of different approxi-

mations, and in our comparison of the contributions of the direct and cascaded fifth

order terms.

Our goal is to identify the parameter space where the power-series expansion of

the local-field-corrected susceptibility, including the total χ(5) (3.27c), is valid. This

defines what we call the “full fifth-order model.” Along the way it will be useful to

also identify the ranges of validity of some other susceptibility models:


• (a) the local-field-corrected χ(1), which is the expression for the total suscep-

tibility neglecting the nonlinear interactions (we refer to this as the “linear

model”);

• (b) the power-series expansion for the local-field-corrected total susceptibility

given up to the third order of nonlinearity (we refer to this expansion as the

“third-order model”);

• (c) the power-series expansion for the local-field-corrected total susceptibility

given up to fifth order of nonlinearity neglecting the cascaded contribution (we

refer to this as the “direct fifth-order model”);

• (d) the full expression for the susceptibility obtained without accounting for

local-field effects, given as [56]

χ =−N |µ|2T2

~

∆T2 − i

1 + ∆2T 22 + |E|2/|E0

s |2(3.32)

(we refer to this model as “full model without LFE”).

We identify the range of parameters over which the models are valid by comparing

them to the full expression for the local-field-corrected total susceptibility,

χ =N |µ|2T2

~

w

T2(∆ + ∆Lw) + i, (3.33)

where the population inversion w is given by (3.1a) (we refer to this model as the


“full model”). In order for a model to be valid for a given range of parameters (the

atomic density and electric field strength), we require that at a given atomic density

and normalized electric-field strength |E|2/|E0s |2 the difference between the full model

and the other model at any detuning is not greater than 3% of the peak value of the

full model. Using this criterion, the ranges over which the models are valid are marked

with colored areas in Fig. 3.1. The full model of Eq. (3.33) is valid everywhere in

|E|2 /|

Es0 |2 N

=0

N (cm-3)

10

1

10-1

10-2

10-3

10-4

10-5

101710161015101410131012

Full model without LFE

Full modelwith LFE

Full fifth-order(with LFE)

Direct fifth-order(without LFE)(with LFE)

Third-order(without LFE)(with LFE)

Linear(without LFE)

(with LFE)

Figure 3.1: The ranges of validity of the various models described in the text for

the total susceptibility of a collection of two-level atoms. The area at the bottom

shows the range of validity of the linear model; local field effects (LFE) are important

in the region on the right but can be ignored on the left. Moving upward on the

plot, the next area shows the range of validity of the third-order model. Again, LFE

can be ignored in the region on the left, and of course this model also accurately

describes the response in the region below it. The next two regions show the ranges

of validity of the direct fifth-order model and the full fifth-order model. In the white

region above these colored regions, the full model of Eq. (3.33) must be used, and

finally in the region at the top of the plot accurate predictions can be obtained by

ignoring LFE as long as the full model of Eq. (3.32) is used.

the plot area. The full model without LFE can be used at higher strengths of the


applied field where the saturation effect is strong, while at small values of the atomic

densities the local-field effects are unimportant and one can neglect them in both the

full model and the power series expansions. The third-order model is valid in the

range where the linear model is valid. The direct and full fifth-order models are valid

in the range of validity of the third model; the full fifth-order model is also valid in

the range of validity of the direct fifth-order model.

Comparing the full fifth-order model to other power-series expansion models, we

conclude that the former has a broader range of validity. As well, we find that the full

fifth-order model gives a more precise description of the local-field-corrected suscepti-

bility than the other power-series expansions in the ranges where all these models are

valid. As an example, we plot the real and imaginary parts of the local-field-corrected

susceptibility given by the different models as the function of the normalized frequency

detuning ∆T2, where the value of T2 is taken at zero atomic density, in Fig. 3.2. We

see that for a given set of parameters all the power-series expansions describe the

total susceptibility fairly well, but the inset reveals that the full fifth-order model

works the best of all, as there is no apparent disagreement between it and the full

model.

We now consider how the presence of the cascaded term affects the size and

frequency dependence of χ(5). In Fig. 3.3 we plot the ratio of the absolute values of

the cascaded and direct terms as the function of the normalized detuning ∆T2 for

several values of the atomic density N within the range of 1× 1014 to 1× 1017 cm−3.


-0.12

-0.08

-0.04

0

0.04

0.08

0.12

-240 -160 -80 0 80 160 240

Re(

χ)

∆T2

(a) Full model [Eq. (3.33)]Full fifth-order

Direct fifth-orderThird-order

-0.002

-0.001

0

0.001

0.002

-38 -37 -36 -35 -34

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

-300 -200 -100 0 100 200 300

Im(χ

)

∆T2

(b) Full model [Eq. (3.33)]Full fifth-order

Direct fifth-orderThird-order

0.126

0.127

0.128

-44 -40 -36 -32 -28

Figure 3.2: Real (a) and imaginary (b) parts of the total susceptibility of a collec-

tion of two-level atoms as functions of the detuning plotted for N = 1016 cm−3 and

|E|2/|E0s |2N=0 = 10−3. The susceptibility is given by different models, as depicted

in the legend.

We take the value of T2 at N = 0 in the normalized frequency detuning. It can be

0

0.25

0.5

0.75

1

1.25

1.5

-2000 -1000 0 1000 2000

R

∆T2

1 × 1017 cm-3

5 × 1016 cm-3

2 × 1016 cm-3 1 × 1016 cm-3

1 × 1015 cm-3

1 × 1014 cm-3

0

0.25

0.5

0.75

1

-40 -20 0 20 40

1 × 1014 cm-3

1 × 1015 cm-3

Figure 3.3: The ratio R = |χ(5)micro|/|χ

(5)dir | of the absolute values of the microscopic

cascaded and direct contributions to the local-field-corrected χ(5) as a function of

the normalized detuning ∆T2 plotted for several values of the atomic density falling

into the range between 1× 1014 and 1× 1017 cm−3. The inset resolves the absolute

values of the ratio for the atomic densities N = 1014 and 1015 cm−3.

seen from Fig. 3.3 that the influence of the local-field effects is two-fold. First, the


local-field effects tend to shift the resonance feature towards lower frequencies by the

amount ∆L. The frequency shift grows linearly with the increase of the density N .

Second, the ratio |χ(5)micro|/|χ

(5)dir| grows with increasing atomic density, as is especially

clear from the inset in Fig. 3.3. This growth saturates at atomic densities higher than

1016 cm−3, because homogeneous collisional broadening comes into play. Clearly, the

cascaded term has a non-negligible contribution to the fifth-order susceptibility.

The result shown in Fig. 3.3, although informative, does not present a complete

picture of the contribution to χ(5) from the cascaded term. We can learn more by

considering the real and imaginary parts of χ(5)dir and χ

(5)micro. These are plotted as

functions of the normalized detuning ∆T2, where T2 is taken at N = 0, for different

values of the atomic density in Fig. 3.4(a), 3.4(c), and 3.4(e). For the purpose of

comparison, we also plot the total fifth-order susceptibility given by Eq. (3.30), which

is the sum of the direct and cascaded contributions, and χ(5)dir [see Fig. 3.4(b), 3.4(d),

and 3.4(f)]. The microscopic cascaded terms make an insignificant contribution until

atomic densities reach on the order of 1013–1014 cm−3 [see Fig. 3.4(a) and 3.4(b)]. As

the atomic density increases, the contribution from the microscopic cascaded term

becomes more pronounced, as one can see from Fig. 3.4(d). The difference becomes

even more significant with further increase of the atomic density, and saturates at the

densities higher than 1016 cm−3.

Taking a careful look at the contribution to χ(5) from the microscopic cascaded

term [better seen in Fig. 3.4(f)], one can see not only a line shape distortion and a


-4

-2

0

2

4

6

8

10

-4 -2 0 2 4

Con

trib

. × |E

s0 |4 N=

0 (1

0-4 e

su)

∆T2

N = 1 × 1013 cm-3

Re(χ(5)dir)

Im(χ(5)dir)

Re(χ(5)micro)

Im(χ(5)micro)(a)

-4

-2

0

2

4

6

8

10

-4 -2 0 2 4

χ(5) ×

|Es0 |4 N

=0

(10-4

esu

)

∆T2

N = 1 × 1013 cm-3

(b)

Re(χ(5)dir)

Im(χ(5)dir)

Re(χ(5))

Im(χ(5))

-20

-10

0

10

20

30

-6 -3 0 3 6

Con

trib

. × |E

s0 |4 N=

0 (1

0-4 e

su)

∆T2

N = 1 × 1014 cm-3

Re(χ(5)dir)

Im(χ(5)dir)

Re(χ(5)micro)

Im(χ(5)micro)

(c)-20

-10

0

10

20

30

-6 -3 0 3 6

χ(5) ×

|Es0 |4 N

=0

(10-4

esu

)

∆T2

N = 1 × 1014 cm-3

(d)

Re(χ(5)dir)

Im(χ(5)dir)

Re(χ(5))

Im(χ(5))

-0.04

-0.02

0

0.02

0.04

0.06

-200 -150 -100 -50 0 50 100 150

Con

trib

. × |E

s0 |4 N=

0 (1

0-4 e

su)

∆T2

N = 1 × 1016 cm-3

Re(χ(5)dir)

Im(χ(5)dir)

Re(χ(5)micro)

Im(χ(5)micro)

(e)-0.04

-0.02

0

0.02

0.04

0.06

-200 -150 -100 -50 0 50 100 150

χ(5) ×

|Es0 |4 N

=0

(10-4

esu

)

∆T2

N = 1 × 1016 cm-3

(f)

Re(χ(5)direct)

Im(χ(5)direct)

Re(χ(5))

Im(χ(5))

Figure 3.4: Real and imaginary parts of the direct and microscopic cascaded

contributions to χ(5), χ(5)dir and χ

(5)micro (a, c, e) and the sum of the contributions,

χ(5)total, plotted together with χ

(5)dir (b, d, f) as functions of the normalized detuning

for different values of the atomic density N .


frequency shift of the maximum, but also a sign change of the imaginary part of χ(5)

in a certain range of detuning. This sign change, of course, cannot be observed in the

total response of the two-level atom, since there is net absorption.

3.4 Experimental Evidence and Applications of Mi-

croscopic Cascading

In the previous sections and in our recent publication [120], we have shown how local-

field effects can act as a mechanism that leads to cascaded microscopic nonlinear

response. In this section, we describe an experiment on separating the microscopic

cascaded contribution to the fifth-order nonlinear susceptibility [121]. This is, to the

best of our knowledge, the first experiment of this kind.

We recall the more general results for the local-field-corrected linear and non-

linear susceptibilities for a centrosymmetric medium from the previous sections (see

Eqs. (3.27)):

χ(1)LFC = Nγ

(1)at L, (3.34a)

χ(3)LFC = Nγ

(3)at |L|2L2, (3.34b)


and

χ(5)LFC = Nγ

(5)at |L|4L2

+24π

10N2(γ

(3)at )2|L|4L3 +

12π

10N2|γ(3)

at |2|L|6L. (3.34c)

Here we added the subscripts “LFC” to the susceptibilities in order to distinguish

them from the effective (total measured) fifth-order susceptibility, which we denote

χ(5)eff , as the latter has an extra, macroscopic (propagational) cascaded term which is

not accounted for in the derivation of Eqs. (3.34).

We recall that the first term on the right-hand side of Eq. (3.34c) is due to the

direct contribution from the fifth-order microscopic hyperpolarizability γ(5)at , while the

two extra terms come from the cascaded contribution of the third-order microscopic

hyperpolarizability γ(3)at . As earlier in this chapter, we refer to the direct term as χ

(5)dir

[Eq. (3.28)], and to the microscopic cascaded terms as χ(5)micro [Eq. (3.29)]. In such a

way, our local-field-corrected fifth-order susceptibility can be represented as

χ(5)LFC = χ

(5)dir + χ

(5)micro. (3.35)

Based on the predictions of Eq. (3.34c), one can conclude that it should be possible to

use microscopic cascading to induce a large cross section for three-photon absorption

(which is proportional to the imaginary part of χ(5)) by making use of a material with

a large value of γ(3)at . The ability to excite three-photon absorption efficiently could


have important implications to nonlinear microscopy and to quantum imaging. This

observation motivated us to undertake a proof-of-principle experiment in an attempt

to isolate the microscopic cascaded contribution to the fifth-order susceptibility.

It is clear from Eq. (3.34c) that χ(5)dir is proportional to the molecular (or atomic)

density N , whereas χ(5)micro is proportional to N2. Hence, in order to experimentally

separate the two contributions to the fifth-order susceptibility, one should measure

the latter as a function of the molecular or atomic density. We have performed such

a measurement in a mixture of carbon disulfide CS2 and fullerene C60, both of which

are highly-nonlinear materials.

Our experimental setup is based on a degenerate four-wave mixing (DFWM)

scheme [122] (see Fig. 3.5) that allows one to separate the effects due to different orders

of nonlinearity. Two beams of equal intensity at 532 nm from a frequency-doubled

M5

M1

M2

M3 M4

P

BSSample

��3��5�

L

Nd:YAG

Figure 3.5: Experimental setup for DFWM. Nd:YAG - 35-ps, 10-Hz, 532-nm

Nd:YAG laser, BS - beam splitter, P - prism (delay arm), M1–M5 - directing mirrors,

L - focusing lens, Sample - 2-mm quartz cell containing a mixture of CS2 and C60.

Nd:YAG laser producing 35-ps pulses were sent into a 2-mm quartz cell containing a

mixture of carbon disulfide (CS2) and fullerene C60, and we observed self-diffraction


phenomena (see the photograph in Fig. 3.5). The first order of diffraction is a con-

sequence of the third-order nonlinear effect, while the second-order diffracted beam

results from the fifth-order nonlinearity.

We measured the intensities of the first- and second-order diffracted beams for

various concentrations of C60 in CS2. In order to correct the nonlinear signals for the

absorption present in the medium, we measured the linear absorption coefficient α and

multiplied our nonlinear signal intensities by the term (αl exp(αl/2)/[1−exp(−αl)])2n,

where l is the length of the nonlinear medium, and 2n+1 is the order of the nonlinear-

ity. We also performed an open-aperture Z-scan measurement [123] to account for the

nonlinear absorption in our samples. Extracting the values of the normalized trans-

mission Tnorm from the Z-scan measurements, we divided our nonlinear intensities by

(Tnorm)2n+1. The third- and fifth-order nonlinear signal intensities, corrected for the

linear and nonlinear absorptions and plotted on a logarithmic scale as functions of

the incident beam intensity, displayed the slopes equal to 3 and 5, respectively (see

Fig. 3.6).

The DFWM experiment yields the absolute values of the nonlinear susceptibilities.

In order to extract these values from the measured intensities of the diffracted beams,

we used the expression [124]

Is = |χ(2n+1)meas |2I2n+1

(n0c

8π

)−2n ( n0c

2πωl

)−2

Ξ(θ), (3.36)

relating the measured intensity Is of the nonlinear signal to the corresponding non-


1

10

100

1000

800 1000 1200 1400

I ou

t, G

W/c

m2

Iin, GW/cm2

1st

diffracted order

2nd

diffracted order

Figure 3.6: Intensities of the first (green circles) and second (red squares)

diffracted orders, corresponding to χ(3) and χ(5) interactions, respectively, as func-

tions of the incident beam intensity plotted on a logarithmic scale. The lines are

the least-square fits by a cubic (green dashed line) and fifth-order (red solid line)

polynomials.

linear susceptibility |χ(2n+1)meas |. Here I1 and I2 are the intensity of an incident beam, c

is the speed of light in vacuo, n0 is the refractive index of the medium, θ is half-angle

between the interacting beams in the experiment, and Ξ(θ) is the phase mismatch

term. We normalized our measured nonlinear susceptibilities to the known value of

χ(3) of the pure CS2, which is 2.2×10−12 esu [56] in order to extract their values from

the experimentally measured intensities.

In Fig. 3.7 we present the typical measured |χ(3)| and |χ(5)eff Ξ(θ)| as functions

of the molar concentration NC60of C60. Here and below in the paper we do not

make an attempt to correct the values of χ(5)eff for the phase mismatch, as we cannot

precisely determine the value of Ξ(θ) in our experiment. The possible sources of error

include lack of precision in measuring the angle between the interacting beams and


imperfections in the geometry of the experiment. We plot the product |χ(5)eff Ξ(θ)|, as

we can extract its values from our experiment using Eq. (3.36) with a good precision.

CS2 and C60 have nonlinear response of opposite sign, which is why both the third-

1.2

1.6

2.0

2.4

0 0.5 1 1.5 2

|χ(3

) | (10

-12 e

su)

C60 (mM/L)

(a)0

0.6

1.2

1.8

2.4

0 0.5 1 1.5 2|χ

(5)

eff

Ξ (θ

)| (10

-22 e

su)

C60 (mM/L)

(b)

Figure 3.7: Typical experimental data for (a) third-order and (b) fifth-order

nonlinear susceptibilities as functions of NC60. The lines represent a least-square fit

with a function (a) linear and (b) quadratic with respect to NC60.

and fifth-order nonlinear susceptibilities in Fig. 3.7 decrease with the increase of

NC60. It is clear from the graphs that |χ(3)| depends on NC60

linearly, whereas |χ(5)eff |

has a quadratic dependence due to cascading. However, as we pointed out earlier

in this section, the total measured fifth-order susceptibility should also include the

macroscopic (propagational) cascaded contribution,

|χ(5)eff | = |χ(5)

LFC + χ(5)macro| = |χ(5)

dir + χ(5)micro + χ(5)

macro|. (3.37)

Both microscopic and macroscopic cascaded effects have a quadratic dependence on

the atomic density [68], and thus separating the contributions of the two cascaded


effects is not straightforward.

In order to resolve the problem of identification of the microscopic cascaded con-

tribution, we solve the driven wave equation [56],

∇2E − ǫ(1)

c2

∂2E

∂t2=

4π

c2

∂2PNL

∂t2, (3.38)

for the direct and microscopic cascaded contributions to |χ(5)eff |, and, separately, for the

macroscopic cascaded contribution. Here PNL(t) = PNL exp(−iωt) = PNL0 exp[i(kr −

ωt)] denotes the nonlinear polarization. The same kind of relationship is valid for the

electric field amplitude: E(t) = E exp(−iωt) = A exp[i(kr−ωt)]. The total fifth-order

nonlinear polarization consists of three contributions, corresponding to the direct and

the two cascaded mechanisms:

P (5) = P(5)dir + P

(5)micro + P (5)

macro. (3.39)

Here we dropped the temporal dependence, as we are dealing with a degenerate

nonlinear effect. Next we will write the separate contributions to the total P (5)

in terms of the electric fields of the interacting waves. After that, we will separately

substitute them into Eq. (3.38) and solve it to obtain the corresponding contributions

to the electric field amplitude, generated by means of the fifth-order nonlinear process.

In order to illustrate how different contributions to the nonlinear signal are gen-

erated, we present a phase matching diagram of our DFWM experiment in Fig. 3.8.


The fundamental electric waves “1” and “2”, propagating at an angle 2θ with respect

12

k1k1

�k2�k2

k3

3 5

k5

��

Figure 3.8: Phase matching diagram for generating first and second diffracted or-

der waves (“3” and “5,” respectively). “1” and “2” are the interacting fundamental

waves. ki are the k-vectors of the corresponding waves.

to each other, interact in the nonlinear medium and produce the diffracted waves

“3” and “5”. The direct and microscopic cascaded contributions to |χ(5)eff | have the

same phase matching condition, as they both are intrinsic properties of the nonlinear

response on the molecular or atomic scale. However, it is not possible to achieve full

phase matching for these terms, as the nonlinear process is degenerate. That is why

it is more informative to present the expression for the wave vector mismatch for the

direct and microscopic cascaded contributions to |χ(5)eff |, which is

∆k5, dir = k5 − (3k1 − 2k2). (3.40)

Here k1 and k2 are the wave vectors of the fundamental interacting beams, and k5

is the wave vector of the generated fifth-order nonlinear signal (see Fig. 3.8). The

corresponding relationship between the nonlinear polarization and the electric fields


is

P(5)dir + P

(5)micro = 10(χ

(5)dir + χ

(5)micro)E

21(E

∗2)

2. (3.41)

Solving Eq. (3.38) for the nonlinear polarization given by Eq. (3.41), we find the

expression

A(5)dir + A

(5)micro =

5π

3n20θ

2(χ

(5)dir + χ

(5)micro)A

31(A

∗2)

2

[

exp

(

i12ωn0

cθ2l

)

− 1

]

(3.42)

for the amplitudes of the electric field, corresponding to the direct and microscopic

contributions to the fifth-order nonlinear signal.

The macroscopic cascaded term, which is a propagational contribution of the

generated third-order nonlinear signal to the fifth-order nonlinear signal, results from

a two-step process with the corresponding wave-vector mismatches:

∆k3 = k3 − (2k1 − k2); (3.43a)

∆k5, macro = k5 − (k3 + k1 − k2). (3.43b)

The nonlinear polarization takes form

P (5)macro = 6χ(3)E1E

∗2E3, (3.44)

where the electric field E3, generated through a χ(3) nonlinear interaction, can be


found from Eq. (3.38):

E(3) =3π

2n20θ

2χ(3)A2

1A∗2

[

exp

(

i4ωn0

cθ2l

)

− 1

]

exp(ik3r). (3.45)

Using Eqs. (3.43b), (3.44), and (3.45), we solve Eq. (3.38) to obtain the relationship

A(5)macro =

35π2

8n40θ

4(χ(3))2A3

1(A∗2)

2

{

1

3

[

exp

(

i12ωn0

cθ2l

)

− 1

]

− 1

2

[

exp

(

i8ωn0

cθ2l

)

− 1

]}

(3.46)

for the macroscopic cascaded contribution to the fifth-order nonlinear signal.

Inspecting Eqs. (3.42) and (3.46), one can see that the amplitudes of the electric

field of the direct and microscopic cascaded contributions have a different depen-

dence on the angle θ and the cell length l, compared to that of the macroscopic

cascaded term. In Fig. 3.9 we plot the absolute values of the angular dependences

of Eqs. (3.42) and (3.46), normalized to unity at θ = 0, as functions of the angle

θ. These dependences characterize the efficiencies of the direct, microscopic, and

macroscopic cascaded contributions to |χ(5)meas|. The normalized efficiency of the di-

rect and microscopic cascaded contributions is shown with the red solid line, and the

efficiency of the macroscopic cascaded contribution is shown with the green dashed

line. As the efficiencies are normalized, we cannot extract the information about the

relative values of the contributions to the total measured |χ(5)eff |. However, the graphs

show approximate positions of the minima and maxima of the efficiencies. It is also


0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6

Effic

iency (

a. u.)

θ (deg)

1 2 34 56

Direct and microMacro

Figure 3.9: Efficiencies of the direct and microscopic cascaded contributions (red

solid line) and the macroscopic cascaded contribution (green dashed line) as func-

tions of the half-angle between the interacting beams. Vertical lines show the ex-

perimental cases.

important that the efficiency of the macroscopic cascaded process decreases much

more rapidly than that of the direct and microscopic cascaded contributions with

the increase of the angle between the interacting beams. Measuring the third- and

fifth-order nonlinear signals at different angles between the interacting beams, it is

possible to discriminate between different contributions to |χ(5)eff |.

The macroscopic cascaded contribution to the total electric field generated by

the fifth-order nonlinear process is proportional to |χ(3)|2. Hence, in our experiment

|χ(5)macro| = Cm|χ(3)|2, where Cm is some parameter independent of NC60

. Neglecting the

direct and microscopic cascaded contributions to the fifth-order susceptibility of pure

CS2, as their values do not change the dependence of |χ(5)eff | on NC60

, we can find Cm

from |χ(5)eff Ξ(θ)|/|χ(3)|2 at NC60

= 0. Then, multiplying the concentration dependence


of |χ(3)|2 by the value of Cm, we find |χ(5)macro Ξ(θ)|. We can estimate |χ(5)

dir + χ(5)micro|

from Eq. (3.37), finding that |χ(5)eff | − |χ(5)

macro| ≤ |χ(5)dir + χ

(5)micro|.

We have measured the nonlinear susceptibilities at four values of the angle between

the interacting beams (marked in Fig. 3.9 with thick vertical lines with the numbers

on top). The results of the measurements are presented in Fig. 3.10 where we plot

the values of |χ(5)eff Ξ(θ)| and |χ(5)

macro Ξ(θ)| as functions of the C60 molar concentration.

For θ ≈ 0.3◦, corresponding to position 1 in Fig. 3.9, we observed no difference

between the |χ(5)eff Ξ(θ)| and |χ(5)

macro Ξ(θ)| [see Fig. 3.10 (a)]. This fact suggests that

for this experimental geometry the macroscopic cascaded contribution to |χ(5)eff | is

much larger than the direct and microscopic cascaded contributions. We repeated

the measurement, increasing the angle to 0.43◦ (position 2 in Fig. 3.9). The resulting

measured values of |χ(5)eff Ξ(θ)| and |χ(5)

macro Ξ(θ)| are represented in Fig. 3.10 (b). One

can see a clear difference between |χ(5)eff Ξ(θ)| and |χ(5)

macro Ξ(θ)|. This means that,

together with the macroscopic cascaded contribution, we observe the presence of

other contributions to |χ(5)eff |, which are the direct and microscopic cascaded terms.

Taking a careful look at position 2 in Fig. 3.9, one can see that, compared to position

1, the curve characterizing the efficiency of the macroscopic cascaded contribution

drops significantly, while the curve describing the efficiency of other two contributions

decreases by a much smaller amount. This explains why we were not capable of

observing the direct and microscopic cascaded contributions in |χ(5)eff | at position 1,


0

0.6

1.2

1.8

2.4

0 0.5 1 1.5 2

|χ| (

10-2

2 esu

)

C60 (mM/L)

(c) 0

0.6

1.2

1.8

2.4

0 0.5 1 1.5 2

|χ| (

10-2

2 esu

)

C60 (mM/L)

(d)

0

2

4

6

8

0 0.5 1 1.5 2

|χ| (

10-2

2 esu

)

C60 (mM/L)

(a)

|χ(5)eff Ξ(θ)|

|χ(5)macro Ξ(θ)|

0

1

2

3

0 0.5 1 1.5 2

|χ| (

10-2

2 esu

)

C60 (mM/L)

(b)

Figure 3.10: Experimentally measured |χ(5)eff Ξ(θ)| and |χ(5)

macro Ξ(θ)| as functions

of NC60. The measurements are done at the angles between the interacting beams

corresponding to (a) position 1 in Fig. 3.9, (b) position 2, (c) position 3, and (d)

position 4. The least-square fits to the experimental data are shown with lines.

but see these contributions in the experimental data taken at position 2.

The angles corresponding to positions 3 and 4 in Fig. 3.9 are in close vicinity to the

minimum of the macroscopic cascading efficiency curve. The corresponding results

are presented in Figs. 3.10 (c) and 3.10 (d). The large difference between |χ(5)eff Ξ(θ)|

and |χ(5)macro Ξ(θ)| indicates that the macroscopic cascaded contribution is not the

dominant contribution to |χ(5)eff |. Observation of positions 3 and 4 in Fig. 3.9 and the


corresponding data in Figs. 43.10 (c) and 3.10 (d) shows a good correlation between

the efficiency curves and the experimental data. Indeed, position 3 corresponds to the

decrease of the macroscopic cascaded contribution efficiency, and position 4 is very

close to the minimum of this curve, while the efficiencies of the direct and macroscopic

cascaded contributions is relatively high at these positions. As a result, we observed

a large difference between |χ(5)eff Ξ(θ)| and |χ(5)

macro Ξ(θ)| in Fig. 3.10 (c), and an even

larger difference in Fig. 3.10 (d).

We next estimate the values of the three contributions to |χ(5)eff | using the data of

Fig. 3.10 (d). The difference of the least-square fits of |χ(5)eff Ξ(θ)| and |χ(5)

macro Ξ(θ)|

contains terms linear and quadratic in NC60. They allow us to estimate |χ(5)

dir Ξ(θ)|

and |χ(5)micro Ξ(θ)|, respectively. Using the data for NC60

= 2 mM/L we extracted

the following values for the three contributions: |χ(5)macro Ξ(θ)| ≈ 9.4 × 10−23 esu, and

|χ(5)dir Ξ(θ)| ≈ |χ(5)

micro Ξ(θ)| ≈ 1.2 × 10−22 esu. These estimates show that under the

conditions of Fig. 3.10 (d) the microscopic cascaded contribution is more significant

than the macroscopic cascaded contribution. The microscopic cascaded term can po-

tentially be larger than the direct term by increasing the molar concentration of C60,

as the former term scales as the square of the concentration. In obtaining the val-

ues presented above, we made several approximations. First, we neglected the direct

and microscopic cascaded contributions to |χ(5)eff | of pure CS2 liquid, because these

contributions do not depend on NC60and we are concerned only with the shape of

the concentration dependence of |χ(5)eff Ξ(θ)|. In addition, we assumed that all three


contributions to |χ(5)eff | have equal complex phases. This assumption allowed us to

estimate the numbers, although there is no fundamental reason to accept the valid-

ity of this assumption. Nonetheless, our primary experimental conclusion, that the

microscopic cascaded contribution makes a significant contribution to |χ(5)eff | remains

valid irrespective of this assumption.

3.5 Conclusions

Performing a theoretical analysis of the local-field-corrected nonlinear susceptibilities

up to the fifth order of nonlinearity, we uncovered the mechanism of microscopic cas-

cading induced purely by local-field effects. Although this effect had been noticed

earlier to occur in third-order nonlinearities, it had been largely overlooked or un-

derestimated in higher-order nonlinear effects. We performed an experimental study

that allowed us to estimate the relative value of the microscopic cascaded contribu-

tion to the total measured fifth-order nonlinear susceptibility. Our study showed that

under certain conditions microscopic cascading can be the dominant effect compared

to other contributions to the fifth-order nonlinearity, including the direct process and

the macroscopic (propagational) cascaded effect.

Microscopic cascading of the sort we analyze here could play an important role

in developing materials with tailored high-order nonlinear response, for instance for

applications in high-resolution imaging. In most of the cases, it is easier to employ

the macroscopic cascaded process, relying on a lower-order nonlinear susceptibility, to


mimic a higher-order nonlinear effect. However, there can be situations in which this

process is unaccessible. For example, for some experimental geometries it may be not

possible to obtain an efficient macroscopic cascading, which we have demonstrated

with some of our experimental data. Moreover, one can think of an experimental case

in which this contribution may be significantly suppressed. That is why it is important

to be aware of an intrinsic property of a high-order nonlinearity, the local-field-induced

microscopic cascaded contribution, which can be potentially made more powerful than

the direct contribution to the high-order nonlinear process. This knowledge should

help one to eventually implement the microscopic cascaded effect for achieving a multi-

photon absorption, which occurs locally, meaning that the macroscopic cascaded effect

is of no use for this phenomenon. Our study presented in this chapter is the first

significant step towards achieving an efficient multiphoton absorption relying on the

local-field-induced microscopic cascading phenomenon.

Theoretical work reported in Chapter 3 has been recently published in Physical

Review A [120]. The experimental work reported in Section 3.4 is to be submitted to

Physical Review Letters [121].

Chapter 4

Enhanced Laser Performance of

Cholesteric Liquid Crystals Doped

with Oligofluorene

4.1 Introduction

Cholesteric liquid crystal structures are produced by mixing a nematic liquid crys-

tal with a chiral additive that causes the nematically-ordered molecules to arrange

themselves into a helical structure. Dye-doped cholesteric liquid crystals (CLCs) are

self-assembling mirrorless distributed-feedback low-threshold laser structures. I have

presented a general overview of this kind of laser structures in Section 1.5. Here I

report my work in the field of dye-doped CLCs.

122

CHAPTER 4. DYE-DOPED CLC LASERS 123

We recall the definition of the emission order parameter from Section 1.5, as this

concept is the basis of our studies reported in the current chapter. When inserted

into a CLC structure, some of organic dyes tend to align with their transition dipole

moments along the local director of the CLC. The degree of alignment is characterized

by the quantity called “order parameter,” defined according to Eq. (1.32) as

Sem =I‖ − I⊥I‖ + 2I⊥

, (4.1)

where I‖ is the fluorescence intensity of the nematic liquid crystal phase doped with

the dye, measured for the radiation with the electric field parallel to the director,

and I⊥ is the fluorescence intensity for the radiation with the electric field polarized

perpendicular to the director. Obviously, the case Sem = 1 corresponds to a perfect

alignment of the dye dipole moment along the liquid crystal director, the case Sem =

−1/2 corresponds to a perfect alignment of the dye dipole moment perpendicular to

the director, and the case Sem = 0 corresponds to an isotropic orientation.

Dyes with high order parameters are believed to have a number of advantages

over the dyes with lower order parameters [82,125]. As most of the molecules’ dipole

moments of such dyes are well-aligned with respect to the local director, the threshold

for lasing oscillations at low-frequency band edge is lower and the efficiency is higher

as compared to the dyes with low order parameter (under the condition that all other

characteristics of the dyes are similar). As there is little or no competition between

the modes corresponding to the low- and high-frequency band edges for the energy


of the pump radiation, the highly oriented dyes can exhibit highly-stable single-mode

oscillations, generating laser radiation only into the mode at the low-frequency edge.

As opposed to the highly oriented dyes, the dyes with lower order parameters can yield

two lasing peaks, corresponding to the low- and high-frequency band edges. It causes

the output radiation to be multi-mode and unstable, as the two frequency modes

compete for the pumping energy. Therefore, to achieve better performance in CLC

lasers, including lower threshold, higher efficiency, and higher stability, it is important

to search for new laser dyes with high order parameters. In reality, one never deals

with two identical dyes with the only difference in their optical properties being the

value of the emission order parameter. That is why comparing two laser dyes with

different order parameters, one has to take into account other optical properties that

can influence laser threshold and efficiency, such as the radiative lifetime, quantum

yield, and the characteristics of the triplet state typical to all organic dyes. The

above characteristics are important if one wants to establish the influence of the

order parameter on the laser threshold and efficiency.

A generation of low-threshold mirrorless distributed-feedback self-assembling lasers

is being actively developed. Since the work published by Kopp et al. [79] brought

light into the origin of lasing in CLC, there were a great number of reports on lasing

in dye-doped [80,82,84,125–134] and undoped [135] CLCs, as well as in liquid crystal

elastomers [136], polymer network devices [137,138], glassy liquid crystals [139], and

cholesteric blue phase [140]. Various methods of tunability in such kind of structures


have been addressed by many researchers: electrical [138], temperature [76, 84, 141],

mechanical [136], optical [142–145], and through concentration [146] and pitch [147]

gradients. Fluorescence behavior in CLC structure has been extensively investigated

both theoretically [81,148,149] and experimentally [81,83,149–152]. For optimization

of the laser performance of dye-doped CLCs, their laser characteristics as functions of

different parameters, such as the CLC structure thickness and dye concentration [127],

temperature [84], excitation rate [129], incident angle and polarization of the pump

radiation [133], and the size of the pump spot [134] have been studied.

DCM dye [4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostryl)-4H-pyran] is

very popular among researchers working in the field of CLC lasers [84, 126–128, 133,

134, 136–138], as this dye can be well incorporated into most of the CLC hosts. On

the other hand, DCM has a low order parameter with a value of around 0.4 [82]. To

the best of our knowledge, there are very few publications reporting an attempt to

find a better dye with higher order parameter for use in CLC lasers [82,125,153].

Here we report a new laser dye, oligofluorene OF2, developed in the laboratory of

Prof. Shaw Horng Chen [154]. We refer to this dye as OF throughout this chapter.

The emission order parameter of OF is 0.60, which is significantly higher than that

of DCM. We perform a comparative study of the new oligofluorene dye and the

commonly used DCM doped into the same CLC material. For both dyes, we find

the optimal concentrations at which their laser performances in CLC are the best.

We demonstrate that oligofluorene has a higher laser output and stability than does


DCM.

In Section 4.2, we describe the procedure of preparing the dye-doped CLC sam-

ples for our studies. In Section 4.3, we present the description of the experimental

setup that we used for laser experiments with CLC structures. In Section 4.4, we

discuss frequency mode competition that influences spatial and temporal stability of

CLC lasers. Section 4.5 is devoted to our experimental results. We summarize our

work in Section 4.6. Sample preparation and their initial quality test was performed

by Simon Wei, a graduate student from Prof. Shaw Horng Chen’s group. All the

laser experiments with dye-doped CLC structures, reported in the current work, were

performed by myself.

4.2 Sample Preparation

We mixed the nematic liquid crystal (ZLI-2244-000, Merck) with the chiral twisting

agent (CB15, Merck) and fluorescent dyes to produce right-handed helical structures.

We filled 22-µm-thick glass cells with the dye-doped mixture. The walls of the cells

were coated with polyimide and rubbed to align the helical axis perpendicular to the

substrates. The cells were prepared in clean room conditions.

The new fluorescent dye OF, an oligomer consisting of a central red-emitting seg-

ment end-capped by tetrafluorenes with aliphatic pendants [154], was used as a highly

oriented candidate for lasing in CLC in our studies. We prepared six samples with

different OF concentrations: 1.00, 1.25, 1.50, 2.00, 2.50, and 3.00 wt. %. For the


purpose of comparison, we prepared the samples doped with 0.50, 0.75, 1.00, 1.25,

and 1.50 wt. % of DCM dye. We also prepared a CLC sample with 1.75 wt. % DCM,

but the clusters of DCM crystals were observed in this sample under a microscope,

which indicates that films with high concentration of DCM encounter phase separa-

tion due to limited solubility of DCM in our CLC host. Nevertheless, the ranges of

the weight percent of the dyes used in our studies cover the optimal dye concentra-

tions for the best performances of both DCM and OF, and these concentrations for

the two dyes are different. Even though the weight percent concentrations of DCM

used in our samples are slightly lower than those of OF, the corresponding molar

concentrations of DCM are almost an order of magnitude higher. For example, 1.00

wt. % of DCM corresponds to a molar concentration 2.64×10−2 M/l, while 1.00 wt. %

of OF corresponds to a molar concentration 2.76 × 10−3 M/l. Observation under a

microscope showed that the CLC structures had large monodomain areas (the areas

without pitch fluctuations). The reflection spectra of the samples revealed an ex-

cellent alignment, displaying sharp interference fringes (see Fig. 4.1). The depressed

baseline on the short-wavelength side of the stop-band in Fig. 4.1 is because of the

dye’s light absorption in the visible region that diminishes the observed reflection

from the dye-doped CLC film.

The pitch length of a CLC structure depends on the concentrations of both chiral

agent and fluorescent dye. We have done a regression calculation to evaluate the

helical twisting power of the chiral agent in the absence of a dye. We then calculated


0

10

20

30

40

50

60

70

500 550 600 650 700

1.25

1.00

0.75

0.50

0.25

0.00

Ref

lect

ance

(%

)

Inte

nsi

ty (

a. u

.)

λ (nm)

(a) ReflectanceFluorescence

Lasing

0

10

20

30

40

50

60

70

550 600 650 700 750

1.25

1.00

0.75

0.50

0.25

0.00

Ref

lect

ance

(%

)

Inte

nsi

ty (

a. u

.)

λ (nm)

(b) ReflectanceFluorescence

Lasing

Figure 4.1: The reflectance (thin solid line) and lasing (bold solid line) spectra of

(a) 1.25 wt. % DCM-doped CLC sample and (b) 3.00 wt. % OF-doped CLC sample

plotted together with the fluorescence spectra of the dyes (dashed line).

how much the presence of 1.00 wt. % of a dye increases the pitch length. Following

these calculations, one can match the low-frequency band edge to the dye’s maximum

fluorescence wavelength with the precision ± 10 nm. As an example, the reflection

and fluorescence spectra of CLCs doped with 1.25 wt. % DCM and 3.00 wt. % OF

are presented in Figure 4.1.

In order to evaluate the emission order parameters of DCM and OF dyes, we

measured the emission intensities polarized parallel and perpendicular to the director

in the nematic liquid crystals doped with the dyes. Then we used Eq. (4.1) to find the

values of the emission order parameters to be 0.36 for DCM and 0.60 for OF. These

values were obtained for 22-µm-thick samples. Additional measurements have shown

that the emission order parameters of 2.5-µm-thick DCM- and OF-doped nematic

liquid crystals are 0.41 and 0.70, respectively. We also measured the absorption

order parameters of the samples and found that their values for DCM- and OF-


doped nematic liquid crystals are 0.43 and 0.76, respectively, and do not change

with the sample thickness. Note that the absorption order parameter can also be

defined by Eq. (4.1), if one replaces the fluorescence intensities by corresponding

intensity absorption coefficients. The values that are significant for our analysis are

the emission order parameters for 22-µm-thick samples (this thickness corresponds to

the thickness of our CLC samples for laser measurements).

4.3 Experimental Setup

Our experimental setup is depicted in Fig. 4.2. We used a frequency-doubled Nd:YAG

laser EKSPLA (Altos), generating 532-nm 35-ps pulses (FWHM) with the 10 Hz rep-

etition rate, as a pump source for the dye-doped CLC structures. The laser produced

Nd:YAG (10 Hz, 35 ps, 532 nm)

Referencedetector

SignaldetectorFocusing

lens

Glass plate

CLCLens

condenser

Computer

λ/2

Linear polarizer

λ/4

Figure 4.2: Experimental Setup.

nearly Gaussian pulses with diameter 3.3 mm FWHM with respect to the intensity

at the position of the focusing lens. A lens of 20 cm was used to focus the laser


beam onto the sample into a spot of 28 µm FWHM. We used a half-wave plate and

a linear polarizer to control the pump energy. In order to ensure maximum absorp-

tion and minimum scattering loss of the pump radiation, we converted the linearly

polarized light exiting the polarizer to left circularly polarized (LCP) light by means

of a quarter-wave plate. The LCP pump radiation penetrated inside a right-handed

CLC sample and got efficiently absorbed by the dye molecules without losses due to

reflection off the CLC structure.

The laser radiation originating from the sample was collected by a lens condenser

comprised of two lenses with the focal lengths 5 cm each. The collected laser radiation

was sent to an energy meter or Ocean Optics spectrometer USB-2000 with 1 nm

resolution. The spectrometer was used to record the laser spectra of the samples. A

sample was mounted on a 3D translation stage (not shown in Fig. 4.2) to provide fine

adjustment of focusing and optimization of alignment.

For measuring the energy of the CLC laser radiation a reference-signal setup

configuration was used. We reflected 10 % of the pump beam by a microscopic glass

plate to the reference energy meter, while the signal energy meter was used to measure

the energy of the CLC laser output. The energy meters were connected to a computer.

Using software, we set a range of acceptable values of the reference energy for each

data point. This way we eliminated the influence of shot-to-shot fluctuations in the

pump laser and significantly increased signal-to-noise ratio in our data.


4.4 Stability and Frequency Mode Competition

Using an electron-multiplied cooled EM-CCD camera (Andor Technologies), we recorded

the intensity distributions of DCM- and OF-doped CLC laser outputs (see Fig. 4.3).

We applied the pump fluence approximately twice the threshold in order to record

the intensity distributions. It can be seen from the figure that DCM has a highly

Figure 4.3: Intensity distribution of the laser output measured using a CCD

camera in (a) 1.00 wt. % DCM-doped CLC and (b) 2.00 wt. % OF-doped CLC.

The dark ring at the bottom of the picture is a camera artifact.

non-uniform intensity distribution in its output, while OF displays a much more uni-

form intensity distribution. The spatial pattern of DCM-doped CLC was found to

change significantly in time, while that of OF remained stable.

We attribute the highly unstable behavior of DCM-doped CLC to the strong de-

gree of competition between the low- and high-energy photonic band edge modes. The

nature of the competition between the frequency modes is as follows. The molecules

of a dye with an order parameter Sem > 0 tend to align with their transition dipole


moments along the local director of a CLC structure. However, there are nonzero

components of the dye molecules’ dipole moments perpendicular to the local director,

as Sem is typically less than 1. This implies that the dye’s emission contributes to

both frequency modes of the CLC situated at the low- and high-energy band edges.

The two band edge frequency modes compete with each other for the use of the pump

energy. When the CLC structure doped with the dye is pumped at the wavelength of

the dye’s absorption, laser generation is most likely to occur at the low-energy band

edge frequency mode, as most of the dye’s emission contributes to that mode. At low

pump energies the low-energy band edge mode suppresses the high-energy edge mode,

as the preferred orientation of the dye’s dipole moments is along the local direction.

As the pump energy grows higher, the competition between the low- and high-energy

band edge frequency modes for the use of the pump energy becomes stronger. At a

certain level of the pump the high-energy edge mode can reach its threshold and the

resulting output spectrum will contain two peaks: a stronger peak corresponding to

the low-energy edge mode, and a weaker peak of the high-energy edge mode.

Typical lasing spectra of our samples are shown in Fig. 4.1, together with the

reflectance of the CLC structures and fluorescence spectra of the dyes. Both DCM-

and OF-doped CLCs displayed lasing at the low-energy band edge. As the order

parameter of DCM is much smaller than that of OF, it is supposed to be relatively

easy to observe the second peak in the lasing spectrum of DCM, corresponding to

the high-energy band edge frequency mode [82, 125, 153]. Nevertheless, the second


peak in the laser outputs of DCM and OF failed to appear, as the lasing degradation

and damage of our CLC host occurred with the increase of the pump fluence before

the high-frequency mode reached its threshold. However, DCM would experience a

stronger competition between the low- and high-energy band edge frequency modes,

regardless whether its lasing spectrum does or does not contain the second peak. We

believe that this competition between the band edge frequency modes caused the

temporal and spatial instabilities that we observed in DCM-doped CLC laser output.

4.5 Lasing Output

We measured the lasing output characteristics of OF- and DCM-doped CLC samples

in two different regimes. The first regime corresponds to the sample position precisely

at the focal point of the lens used to focus the pump radiation. In this regime we

observed laser output of a sample in the form of a single spot corresponding to the

single transverse fundamental spatial mode (see Fig. 4.4(a)). Due to this reason,

we call this regime “transverse single-mode.” We positioned the sample precisely

to the focal point of the lens by translating it laterally on the micrometer stage

while measuring the laser output at a very low pump level (slightly higher than the

threshold). This way, we optimized for the maximum output energy, which helped us

to ensure that the sample is precisely at the focus.

In the second regime we defocused the pump radiation by longitudinally trans-

lating the sample 14 mm away from the focal point of the lens. In this case, we


observed a ring pattern at the sample’s output, corresponding to generation of sev-

eral transverse spatial modes (see Fig. 4.4(b)). We call the second regime “transverse

multi-mode.” The pump spot diameters in transverse single-mode and multi-mode

regimes were 28 and 230 µm FWHM, respectively. We have done a comparative study

of the laser performances of DCM- and OF-doped CLCs in both regimes.

Figure 4.4: (a) A photograph of a single transverse mode observed in the lasing

output of the 2.00 wt. % OF-doped CLC in transverse single-mode regime. (b)

A photograph of a ring pattern observed in the lasing output of the 2.00 wt. %

OF-doped CLC in transverse multi-mode regime.

4.5.1 Transverse single-mode regime

Laser output characteristics in transverse single-mode regime, in which the samples

were positioned precisely at the focal point of the lens focusing the pump radiation, are

presented in Fig. 4.5. The way we obtained the data points in Fig. 4.5 is by averaging

over 30 measurements for each pump energy setting and evaluating the standard


0

10

20

30

40

50

0.50.40.30.20.10.0

DC

M o

utp

ut

ener

gy

(n

J)

Pump energy (µJ)

(a)0.50 wt. %0.75 wt. %1.00 wt. %1.25 wt. %1.50 wt. %

0

10

20

30

40

50

1.81.51.20.90.60.30.0

OF

ou

tpu

t en

erg

y (

nJ)

Pump energy (µJ)

(b)1.00 wt. %1.25 wt. %1.50 wt. %2.00 wt. %2.50 wt. %3.00 wt. %

Figure 4.5: Laser output energy plotted as a function of the incident pump energy

of (a) DCM- and (b) OF-doped CLC samples in transverse single-mode regime.

deviation. Different curves correspond to different concentrations of DCM and OF,

as reflected in the legends. The ranges of the pump energies, shown on the X-axes of

the graphs, are different, as the output characteristics of DCM-doped CLCs saturated

much more rapidly with increasing the pump energy. The Y -axes of the graphs,

representing the output energy of the samples, have the same scale, and it is obvious

that OF produces 5 times more output energy in transverse single-mode regime, as

compared to DCM. The output vs. pump energy characteristics in transverse single-

mode regime were reproducible within 10%. The standard deviation of the measured

output energy was less than 10%. The lasing threshold fluences of all DCM and OF

samples are approximately the same and are around 7 mJ/cm2. The slope efficiencies

derived from the linear parts of the output characteristics as the ratios of the output

energy changes to the changes in the incident pump energy are presented in Fig. 4.6.

It is clear from Figs. 4.5 and 4.6 that there is an optimal concentration for each laser


8.0

7.0

6.0

5.0

4.0

3.0

2.03.002.502.001.501.000.50

η (%

)

Dye (wt. %)

DCMOF

Figure 4.6: Slope efficiency of the laser output of DCM- (squares) and OF-doped

CLC samples (circles) as a function of the dye weight percent in transverse single-

mode regime.

dye at which the CLC laser demonstrates the top performance, yielding the highest

output energy and slope efficiency. At the dyes concentrations lower than the optima,

there is not enough dye molecules in CLC to produce much energy and efficiency. At

high dye concentrations the pump radiation gets absorbed within several microns from

the front surface of the sample and cannot get inside the CLC structure far enough

for the dye molecules all through the length of the CLC to be excited. Besides, in case

of high dye concentrations the dye molecules are spaced so closely that the parasitic

reabsorption and triplet quenching effects characteristic to all organic dyes [155] get

much stronger. The maximum slope efficiency achievable in transverse single-mode

regime was around 5% for both DCM and OF dyes. It can be seen from Fig. 4.6 that

the best laser performance was achieved with 1.25 wt. % DCM and 2.00 wt. % OF


samples. The slope efficiency data were reproducible and accurate within 10%, which

is reflected in the size of the error bars in Fig. 4.6. Within this study, we report the

slope efficiency values based on the measurement of the CLC laser output collected

only in one direction.

4.5.2 Transverse multi-mode regime

Laser output characteristics measured in transverse multi-mode regime, correspond-

ing to the position of the samples 14 mm away from the focal spot of the lens, are

shown in Fig. 4.7. The reproducibility of the data obtained in transverse multi-mode

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 0 10 20 30 40 50 60 70 80 90

DC

M o

utp

ut

ener

gy (

µJ)

Pump energy (µJ)

(a) 0.50 wt. %0.75 wt. %1.00 wt. %1.25 wt. %1.50 wt. %

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 0 20 40 60 80 100 120 140

OF

outp

ut

ener

gy (

µJ)

Pump energy (µJ)

(b)

1.00 wt. %1.25 wt. %1.50 wt. %2.00 wt. %2.50 wt. %3.00 wt. %

Figure 4.7: Laser output energy plotted as a function of the incident pump energy

of (a) DCM- and (b) OF-doped CLC samples in transverse multi-mode regime.

regime is within 20%, while the standard deviation of the output energy is around

10%. The maximum output energy of OF in transverse multi-mode regime was mea-

sured to be 1.6 times greater than that of DCM. The lasing threshold fluences of all

OF and DCM samples were around 0.7 mJ/cm2, which is an order of magnitude lower


than that in transverse single-mode regime. The reason for the higher threshold in

transverse single-mode regime can be that the pump is focused so tightly that the

radiation with such a small beam diameter cannot efficiently pump the transverse

fundamental spatial mode. This makes it more difficult to achieve the threshold on

that mode, and makes it impossible for higher-order spatial modes to appear at the

output of the CLC laser. The latter is good when it is crucial to obtain a single

transverse fundamental spatial mode in the CLC output.

The slope efficiencies of the samples in transverse multi-mode regime are shown

in Fig. 4.8. Unlike in the transverse single-mode regime, where the slope efficiencies

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.03.002.502.001.501.000.50

η (%

)

Dye (wt. %)

DCMOF

Figure 4.8: Slope efficiency of the laser output of DCM- (squares) and OF-doped

CLC samples (circles) as a function of the dye weight percent in transverse multi-

mode regime.

of DCM and OF were comparable, in the transverse multi-mode regime OF displayed

almost twice larger maximum slope efficiency. A possible explanation to this is as


follows. In the transverse multi-mode regime there are many spatial modes in the laser

output. They all contribute to the total laser output energy, and, therefore, influence

the overall slope efficiencies of the output characteristics. As it has been shown in the

transverse single-mode regime, the saturation of the output energy with the increase

of the pump energy occurs much more rapidly in DCM- than in OF-doped CLCs. In

the transverse multi-mode regime, there are many modes contributing to the total

output of DCM, but each of those modes saturates more rapidly than a similar mode

of an OF-doped CLC. That is why the overall slope efficiency of DCM-doped CLCs

is significantly lower in the transverse multi-mode regime.

Based on the above picture, we can also explain the reason why in the transverse

single-mode regime OF-doped CLCs produce 5 times higher maximum output energy

as compared to DCM-doped CLCs, while in the transverse multi-mode regime they

produce only 1.6 times higher maximum output energy. In the transverse single-mode

regime we measure the output only of the fundamental spatial mode. That is why

the maximum output energies of the dye-doped CLCs are limited by the saturation

of that mode. In the multimode regime, the pump spot is an order of magnitude

larger, and it efficiently pumps several spatial modes. As the intensity distribution

at the pump spot on the sample is Gaussian, the first mode to lase would be the

fundamental mode. The other modes switch on as we increase the pump energy, so

that the intensity at the edge of the pump spot is high enough for them to meet

the threshold conditions. As we increase the pump energy, more and more spatial


modes appear in the laser output, and DCM-doped CLCs saturate not as rapidly as

they would if there were only one fundamental mode in the output. That is why

the difference in the maximum output energy between DCM- and OF-doped CLCs is

smaller in the transverse multi-mode regime.

Comparing the laser performances of DCM- and OF-doped CLC samples, we

found that the laser thresholds are similar for all samples. All DCM- and OF-doped

CLC display the thresholds 7 mJ/cm2 and 0.7 mJ/cm2 in the transverse single-mode

and multi-mode regimes, respectively. The slope efficiencies of DCM- and OF-doped

samples are similar in the transverse single-mode regime. In the transverse multi-

mode regime OF-doped samples display the slope efficiency almost twice higher than

that of DCM-doped samples. The maximum laser output obtained with OF-doped

CLCs was found to be 5 times greater than that obtained with DCM-doped CLCs

in the transverse single-mode regime, and 1.6 times greater in multi-mode regime.

In addition, the spectral purity and the temporal and spatial stability of the laser

output of OF-doped CLCs is much higher than that of DCM-doped CLCs. Based

on the above results, one can conclude that OF is a better choice for lasing in CLC

structures.

4.5.3 Laser Output Degradation Issues in CLCs

Working with dye-doped CLC in liquid phase requires extra care. Even a slightest

mechanical stress that one accidentally applies to a CLC sample can cause the loss


of alignment, appearance of multi-domain regions, and change in the pitch and the

lasing wavelength. Besides, as the host is in liquid state, the period of the structure

is sensitive to the heating from the pump radiation, and degradation of the CLC laser

output, caused by the heating, can occur. Even though glassy liquid crystal hosts are

more robust, it is the CLCs in liquid state that allow to achieve tunability of the laser

wavelength. Therefore, it is important to investigate CLC lasers with both liquid and

glassy hosts. The experiments with glassy liquid crystal hosts doped with OF dye

have been recently performed in Prof. Chen’s group [156].

We observed a difference in the behavior of DCM- and OF-doped CLC lasers.

For example, in the transverse single-mode regime at the pump energies in the range

between 200 and 500 nJ, we observed a significant degradation of laser output with

time in DCM-doped CLCs, while OF-doped CLCs displayed stable lasing in this range

of pump energies.

4.6 Conclusions

We have performed a detailed comparative study of the laser output characteristics

of CLC structures doped with a commonly used laser dye, DCM, with the emission

order parameter 0.36 and with a new laser dye, OF, with the order parameter 0.6.

The study of the laser spectra showed that only a single laser peak, corresponding

to the low-energy band edge frequency mode, can be observed in the output of all

DCM and OF samples. We expected to observe a second peak, corresponding to


the high-energy band edge mode in the output spectra of DCM-doped CLCs, but it

appears that the threshold for the second frequency mode is higher than the damage

threshold of our CLC host.

OF-doped CLCs displayed a much higher temporal and spatial stability in the

output radiation. We attribute this to the higher value of OF’s order parameter

that prevents strong competition between the high- and low-energy edge frequency

modes. Strong competition between the modes, taking place in case of a dye with a

low order parameter, degrades the laser performance, creating temporal and spatial

instabilities, which we observed with DCM-doped CLCs.

We measured the laser output characteristics of DCM- and OF-doped CLCs in

two different regimes corresponding to generating a single fundamental spatial mode

and a multi-mode ring pattern. This study itself is interesting, as, to the best of

our knowledge, spatial transverse single-mode and multi-mode regimes have not been

discussed in CLC structures before. The value of the transverse single-mode regime is

that one can obtain a single fundamental spatial mode at low pump energy. Transverse

multi-mode regime is interesting because it yields much higher output energy. The

way one can control the number of the spatial modes appearing in the output is

through changing the size of the pump spot in the sample. We found that the laser

thresholds and the slope efficiencies of DCM- and OF-doped CLCs are similar in

the transverse single-mode regime, but OF-doped CLCs produce maximum output

energy 5 times greater. The laser thresholds of the samples doped with both dyes in


the transverse multi-mode regime are similar, but the slope efficiency of the OF-doped

samples is almost twice higher.

The results of our study invite the conclusion that OF is an excellent laser dye

for use in CLC structures, demonstrating a better laser performance in many aspects

than that of the popular dye DCM.

This work has been published in the Journal of the Optical Society of America

B [157]. New experimental work on characterization of OF-doped robust glassy CLCs

have been recently performed in Prof. Shaw Horng Chen’s group [156]. Several other

experiments related to dye-doped liquid and glassy CLC structures are currently in

progress.

Chapter 5

Optical Activity in Diffraction

from a Planar Array of Achiral

Nanoparticles

5.1 Introduction

Chiral objects occur in two mirror-image forms (enantiomers) that cannot be super-

imposed with each other by proper rotations [158]. Chirality is usually associated

with molecular structure and leads to optical-activity effects, which arise from dif-

ferent interaction of chiral molecules with left- and right-hand circularly-polarized

light [159, 160]. Conventional optical-activity effects, such as circular dichroism and

polarization (azimuth) rotation, arise from “molecular” chirality and occur in isotropic

144

CHAPTER 5. ARTIFICIAL CHIRAL STRUCTURES 145

bulk liquids (e. g., sugar solutions) and molecular crystals. Because of their molecular

origin, these effects are proportional to the density of chiral molecules and build up

as light traverses the chiral medium. Optical-activity effects can also arise from a

chiral arrangement of achiral objects, e. g., the arrangement of silicon and oxygen

atoms in a unit cell of crystalline quartz. Such “structural” chirality relies on the

stability of the atomic arrangement and vanishes when the crystal is melted or dis-

solved. Both molecular and structural chirality thus arise from the three-dimensional

(3D) nature of the material. It is also possible that materials with neither molecu-

lar nor structural chirality give rise to optical-activity effects. This is the case if an

experiment is performed where the setup itself is chiral, i. e., it is defined by three

non-coplanar vectors with a given handedness. Such effects are known in light scatter-

ing from anisotropic molecules or angular-momentum-aligned atoms and in nonlinear

optics [161–165]. Note also that molecular chirality implies structural chirality, and

structural chirality implies chirality of the experimental setup. Therefore, separation

of the three different mechanisms may be difficult.

Recent nanofabrication techniques have made it possible to prepare samples with

so-called planar or 2D chirality [85, 166]. Such samples are usually 2D arrays with

a sub-wavelength period and consist of nanoparticles that cannot be brought into

congruence with their mirror image by in-plane rotations or translations. The sam-

ples, therefore, possess a sense of twist, which is different when viewed from the front

and back sides of the sample. This peculiarity has invoked controversies of possible


violation of time-reversal symmetry and nonreciprocity of the light-matter interac-

tion in 2D chiral media [89, 96, 97]. However, it is now well established that planar

arrays of nanoparticles lead to optical activity similar to that of conventional chiral

media [90], because of the front–back asymmetry brought about by the substrate [90]

or other vertical structure [167], which effectively turns a 2D sample into a 3D sam-

ple. However, it is now well-established that planar arrays of nanoparticles lead to

optical activity similar to that of conventional chiral media [90]. The reason is that

a real 2D chiral sample can be seen as a 3D sample because of the substrate [90] or

other vertical structure [167]. As a consequence, conventional circular dichroism and

polarization rotation in transmission are forbidden in strictly 2D samples, such as

free standing nanogratings [90].

In contrast to the conventional optical activity, which is thus forbidden, polarization-

sensitive diffraction from 2D chiral samples can occur. Different diffraction patterns

for left- and right-hand circularly-polarized light have been observed for planar square

gratings consisting of particles with four-fold rotation axis and no reflection symme-

try [97]. Following the above classification for bulk media, this result can be seen

to arise from 2D molecular chirality because of the particular sense of twist of the

individual 2D nanoparticles. Similarly, one can also introduce 2D structural chiral-

ity, when the sense of twist of the 2D grating arises from the mutual arrangement

of achiral nanoparticles. An attempt has been made to separate contributions from

the molecular and structural 2D chirality to the polarization effects in diffraction


experiments by varying the direction of polarization of incident light [85]. However,

no reference sample with pure structural chirality was used. Such a sample would

have to consist of achiral nanoparticles arranged in a chiral grating with no mirror

symmetry.

In this study, we report the observation of polarization changes in diffraction from

a planar grating that consists of achiral crosses whose mutual orientations make the

overall sample 2D chiral. This is, to the best of our knowledge, the first systematic

study of diffraction from planar structures with pure structural chirality. We find

that the polarization changes from such samples are significant, in fact, comparable

to those observed from reference samples where the individual particles are chiral.

This is surprising, as the origin of chirality in the two samples is different, and the

interparticle coupling would be crucial for observing optical activity in transmission

experiments with structurally chiral samples. It is therefore quite natural to expect

that the effects may be quantitatively different for samples with chirality of different

origins; we show, however, that in the diffraction experiments this is not the case. Our

results therefore suggest that the two types of sample chirality cannot be separated in

diffraction experiments. Our conclusion is further supported by a simple model which

describes the polarization of the diffracted wave in terms of light scattering from a

strictly 2D array of nanoparticles. In this model, the grating only enhances scattering

into a given diffraction order through constructive interference, as the model involves

no interparticle coupling.


5.2 Experiment

Our sample, fabricated utilizing electron-beam lithography, contains different patterns

of nanoparticles on a fused silica substrate. In the direction perpendicular to the

sample plane, the particles consist of a 3 nm Cr adhesion layer and 30 nm of Au. The

entire structure is coated by 20 nm protective SiO2 layer. There are eight different

nanoparticle patterns (Fig. 5.1). Patterns 1 and 2 are achiral: pattern 1 is a square

Detector

Rotatable table

He-Ne laserL1M1

M2

SampleQWP

Polarizer L2

1 3 5 7

2 4 6 8

Figure 5.1: Experimental setup and sample layout.

lattice of achiral crosses with the legs oriented along the lattice axes, while pattern 2

is a lattice of crosses tilted at 45◦ with respect to the lattice axes. Patterns 3 and 4

consist of achiral crosses tilted at +27.5◦ and −27.5◦ with respect to the lattice axes,

respectively. The ±27.5◦ tilt results in pure structural chirality of the patterns, as

the individual particles are achiral. Patterns 5, 6, 7, and 8 contain chiral gammadions

and propellers, and, therefore, possess molecular chirality. The chiral patterns in each


pair (3, 4), (5, 6), and (7, 8) are enantiomeric forms (mirror images) of each other.

The period in all gratings is 800 nm; each pattern is a 1 mm × 1 mm square.

We illuminated the sample with a linearly polarized light from a 633-nm He-Ne

laser at normal incidence (see Fig. 5.1). The polarization state of the incident light was

controlled by a half-wave plate and a linear polarizer (not shown in the figure). We

measured the polarization rotation and ellipticity of the first-order diffracted beams

from all eight patterns for p- and s-polarized incident light. (We call the incident light

p-polarized if its electric field vector lies in the plane perpendicular to the sample and

containing the diffracted beam, and s-polarized if the electric field is perpendicular

to this plane.) Due to the astigmatism of the electron beam writing, the samples

can exhibit some anisotropy that affects the polarization state of the diffracted light.

To remove the effects of the residual anisotropy of the structures, we analyzed the

polarization states of four equivalent first-order diffracted beams corresponding to the

four equivalent azimuthal orientations of the samples, and averaged the results. The

spread in the data collected from the four equivalent first-order diffracted beams does

not exceed 3◦ for patterns 1, 5, 6, 7, and 8, but is about 10◦ for patterns 2, 3, and

4, as the lattice axes for the latter patterns have a nonzero angle with respect to the

directions of the electron beam writing. We repeated each measurement three times

and found that the random errors in our measurements are smaller than the size of

the datapoint in the graphs. However, the precision of the polarizer and the quarter-

wave plate rotation is within 1◦, which introduces some uncertainty into establishing


the “zero level” of ellipticity and polarization rotation. This zero level is set by the

achiral patterns 1 and 2 and we have removed this offset from the other results.

5.3 Experimental Data

The polarization rotation and ellipticity of the diffracted beam are presented in

Fig. 5.2. The achiral patterns 1 and 2 do not change the polarization state of the

-6

-4

-2

0

2

1 2 3 4 5 6 7 8

Elli

ptic

ity, d

eg

# of pattern

(b)

-4

0

4

8

Azi

mut

h ro

tatio

n, d

eg (a)p-polarizations-polarization

Figure 5.2: (a) Polarization azimuth rotation and (b) ellipticity of light in diffrac-

tion from the planar structures. Data for p- and s-polarized incident light are shown

with squares and circles, respectively. The results for the two enantiomerically op-

posite forms are connected by the arrows. The offset of the “zero level” set by the

achiral patterns 1 and 2 is within 1◦. It has been shifted to coincide with the zero

level of the graphs for both polarization rotation and ellipticity.

diffracted light. The chiral patterns 3–8, on the other hand, display clear deviations

of the polarization rotation and ellipticity with respect to the zero level established

by the patterns 1 and 2. The patterns that are enantiomeric forms of each other


demonstrate polarization effects equal in magnitude and opposite in sign, which is a

manifestation of chirality. The polarization effects reverse sign as the linear polar-

ization of the incident light changes from p- to s-state. The strongest polarization

changes are observed from the pair of patterns (5, 6), which contain chiral gam-

madions. The effect observed from the pair of patterns (3, 4) with pure structural

chirality, however, is almost as large. Patterns 7 and 8, consisting of chiral propellers,

on the other hand, display very weak polarization changes.

5.4 Wire Model

The origin of the polarization effects in patterns 3 and 4 possessing pure structural

chirality can be explained by a simple “wire model” (inspired by [85]) where the only

possible source of chirality is the orientation of the achiral particles with respect to the

square lattice. Let us assume that a plane monochromatic wave with the frequency ω

is incident on a patterned surface with the period a (for convenience, we do not show

the frequency dependence). We consider the linear response of the surface current

density j(ρ) to the electric field Ei(ρ) of the incident electromagnetic wave, where ρ

is a 2D position vector in the plane of the sample surface:

ji(ρ) = κij(ρ)Eij(ρ). (5.1)


We can treat the linear surface response tensor↔κ(ρ) as a 2 × 2 matrix, as Ei(ρ) lies

in the surface plane in the case of normal incidence. In order to find the electric field

in the far zone that is emitted by the current distribution j(ρ), we need to represent

j(ρ) as a spatial Fourier series. Since Ei(ρ) is homogeneous in the surface plane, the

problem reduces to a spatial Fourier representation of↔κ(ρ). As the sample is periodic

in both main directions of the square lattice (which we call X and Y directions), we

can write it down as

κij(ρ) =∑

αβ

κ(α, β)ij exp

[

i2π

a(αρx + βρy)

]

, (5.2)

where

κ(α, β)ij =

1

S

∫

S

κij(ρ) exp

[

−i2π

a(αρx + βρy)

]

dρ, (5.3)

S is the area of the unit cell. The zeroth order of the expansion (5.3), κ(0,0)ij ≡ κij,

describes the reflected wave. The term with κ(1,0)ij describes the first order of diffraction

in the direction of positive x, i. e., one of the four first-order diffraction maxima in

the reflection geometry.

We model each “molecule” with a cross made of two wires of the length 2d each;

the cross is rotated counter-clockwise by an angle α with respect to the XY co-

ordinate system of the square lattice (see Fig. 5.3(a)). As evident from standard

diffraction theory, the waves emitted by different unit cells are coherently added only

in the directions of the diffraction maxima determined by the lattice period a. We


�

Y

X

(b)(a)

Zdiffracted

light

jx X

j x sin�

�

�

E p

dif

Es

dif

jy

Figure 5.3: Scattering from a tilted cross. (a) Top view of the cross. (b) Side view of the scattering

geometry.

consider the scattering of light from the pattern in the direction of the first diffraction

maximum (1, 0), which lies in the XZ-plane and has the angle of diffraction ξ (see

Fig. 5.3(b)). The electric field Edif in the far zone in that direction is determined by

the respective spatial Fourier harmonic of j(ρ), which can be written as

j(1,0)m = κ(1,0)

mn Ein. (5.4)

A straightforward calculation yields the result

↔

κ(1,0) = σa

π

β cos α + γ sin α β sin α − γ cos α

β sin α − γ cos α βsin2α

cos α+ γ

cos2α

sin α

. (5.5)

Here σ is the specific linear conductivity of the wires, β = sin[(πd/a) cos α], and

γ = sin[(πd/a) sin α]. The s-component of diffracted light, Edifs , is collinear with

the Y axis, while its p-component, Edifp , makes the angle ξ with the X axis (see

Fig. 5.3). Therefore, we can find the components Edifs and Edif

p from Edifs = Cj

(1,0)y


and Edifp = Cj

(1,0)x sin ξ, where C is a constant that is identical for both Edif

s and Edifp .

These results then directly yield the polarization rotation.

In Fig. 5.4 we show the theoretical polarization rotation of the diffracted electric

field as a function of the tilt angle α of the wire cross with respect to the lattice axes

for the p- and s-polarized incident light. The graphs are plotted for the leg length

-8

-6

-4

-2

0

2

4

6

-45 -30 -15 0 15 30 45

Pola

riza

tion

rota

tion

(deg

)

α (deg)

#3, p

#3, s

#4, p

#4, s

#7, p

#7, s#8, p

#8, s#1, p

#1, s

#2, s and pp-polarizations-polarization

Figure 5.4: Polarization azimuth rotation of the diffracted light as a function of

the tilt angle α of the crosses with respect to the lattice axes, calculated using the

wire model. The solid and dashed curves correspond to the p- and s-polarization of

the incident light, respectively.

of the crosses 2d = 500 nm and the lattice period a = 800 nm, which approximately

corresponds to the geometry of our samples. The case of α = ±27.5◦ corresponds to

our patterns 3 and 4 (shown as datapoints on the same graph). It is clear that the

predictions of the wire model are in good agreement with the respective experimen-

tal data. In particular, the wire model predicts a smaller effect for the s-polarized

incident light compared to that for the p-polarized incident light, which we observed

experimentally. Patterns 7 and 8, which are chiral propellers, show a very weak polar-

ization rotation and elliptization. These patterns resemble achiral crosses tilted with


respect to the lattice axes by a very small angle. From Fig. 5.1 we can estimate the

tilting angle to be about 10◦, with the tilting directions of the patterns 7 and 8 being

the same as those of patterns 4 and 3, respectively. We also plotted the datapoints

corresponding to patterns 7 and 8 in Fig. 5.4. The predictions of our wire model for

this tilting angle agree well with our experimental data.

The wire model illustrates that the 2D structural chirality manifests itself in the

optical activity of diffracted light, similar to that from pure 2D molecular chirality.

This occurs even for highly symmetric normal incidence. For achiral particles, the

sample is chiral whenever a distinct symmetry direction of each particle is tilted with

respect to the lattice axes of the sample. For chiral particles the sample is always

chiral, because the individual particles possess no symmetry direction with which

to compare the lattice axes. Consequently, attempts to separate structural from

molecular chirality in a sample consisting of chiral particles would be meaningless.

The similar polarization effects for both molecular and structural chirality can also

be understood as manifestations of the chirality of the diffraction setup. Such chirality

arises from the shape and orientation of a single nanoparticle in the setup, with the

lattice only defining the directions of non-zero diffraction orders. The origin of the

chirality of the setup can also be understood from a crystallographic perspective.

Indeed, the 2D point symmetry group of both kinds of our chiral samples is p4,

which lacks the in-plane mirror-reflection operation. Although the normal incidence

geometry in our experiment is as symmetric as it can be, the diffraction direction is


off-normal as determined by the symmetry of the lattice. These factors give rise to the

left–right asymmetry required for observing the polarization effects. It is important

to keep in mind, however, that these symmetry-based considerations cannot give

quantitative predictions, as they are not based on a detailed microscopic analysis and,

in particular, do not include any assumptions about interparticle coupling. Thus, our

experimental results and their analysis help to build a more detailed understanding

of the polarization effects.

5.5 Conclusions

In conclusion, we have reported the observation of polarization rotation and elliptiza-

tion of light diffracted from planar arrays consisting of either chiral or achiral nanopar-

ticles. We have shown that, in both cases, the polarization changes are comparable;

they are present even for non-mirror-symmetric patterns consisting of achiral parti-

cles (i. e., possessing pure structural chirality). Our experimental data are in good

agreement with the predictions of a simple wire model that describes light scattering

from a planar achiral particle and includes no interparticle coupling. Our analysis

of the data and the model suggests that diffraction experiments cannot distinguish

between the polarization effects arising due to molecular and structural chirality of

the samples. We believe that our study reported here clarifies the misconceptions re-

lated to separating molecular and structural chirality in diffraction experiments. Our

findings also demonstrate that, in practical applications, patterns with pure struc-


tural chirality can be as efficient as those consisting of chiral particles, while having

less pattern complexity and thus being easier to fabricate. Artificial planar struc-

tures with pure structural chirality can thus be considered as a whole new promising

class of materials to be used in polarization control devices. This conclusion is not

limited to metal-based structures considered in the present paper, but is also true for

all-dielectric planar chiral structures [168], which may be promising because of lower

losses and which will be a subject of a separate study.

The work reported in this chapter is a collaboration between Prof. Boyd’s group,

Dr. Sergei Volkov, and the groups of Prof. Martti Kauranen, Prof. Yuri Svirko, and

Prof. Jari Turunen. It is submitted to Physical Review Letters [169].

Chapter 6

Summary

My Ph. D. dissertation, entitled “Local-Field Effects and Nanostructuring for Con-

trolling Optical Properties and Enabling Novel Optical Phenomena,” consists of four

major projects. Below I summarize the projects and my personal contributions.

1. “Composite Laser Materials”

Within this project, I have performed both theoretical and experimental inves-

tigations of the modification of laser properties of composite materials. First,

Prof. Boyd and I have performed a proof-of-principle theoretical study of the

modification of basic laser parameters by local-field effects, treating a compos-

ite laser gain medium as a quasi-homogeneous material. We have shown that

significant changes to the basic laser parameters can be achieved.

However, more precise and sophisticated models are needed for separate com-

posite geometries. I have developed such models for Maxwell Garnett composite

158

CHAPTER 6. SUMMARY 159

geometry with the resonance in inclusions and for layered composite geometry.

Following my recipe, one can numerically solve the problem for more complex

composite structures, such as Bruggeman geometry, where local field is non-

uniform. An experimental validation of the theory will be performed by other

graduate students from Prof. Boyd’s research group.

The experimental part of the project involved the measurement of the radiative

lifetime of Nd:YAG nanoparticles suspended in different liquids. I have experi-

mentally demonstrated that the local-field effects can cause a significant change

in the radiative lifetime of composite materials compared to that of the bulk

materials. My experimental results obey the real-cavity model of local field.

This conclusion should help to resolve a controversy in the literature regarding

the influence of local-field effects on the radiative lifetime of liquid suspensions

of nanoparticles.

This work is a collaboration with Prof. Peter Milonni. It is described in de-

tail in Chapter 2 of the current thesis. The project gave a material for three

publications: [24,105,116].

2. “Microscopic Cascading Induced by Local-Field Effects”

Within this project, I have theoretically predicted the existence of microscopic

cascaded contribution from the third-order hyperpolarizability to the fifth-order

nonlinear susceptibility, induced by local-field effects. This effect resembles the

macroscopic, propagational cascading, in which a two-step third-order nonlinear


effect contributes to the fifth-order susceptibility. However, the microscopic

cascading has a local nature and does not require propagation.

Performing a degenerate four-wave mixing experiment, I have successfully iden-

tified the microscopic cascaded effect and have shown that under certain con-

ditions it can potentially be the dominant contribution to the total measured

fifth-order nonlinear susceptibility. The reason why the microscopic cascad-

ing effect is important is because it has a potential application in the field of

quantum optics where N -photon absorbing materials are needed.

This work is a collaboration with Prof. John Sipe and my colleague Heedeuk

Shin, a graduate student from Prof. Boyd’s group. It is covered in Chapter 3.

The project resulted in two publications: [120,121].

3. “Cholesteric liquid crystal laser doped with oligofluorene”

This project is a collaboration between Prof. Shaw Horng Chen’s research group,

Dr. Svetlana Lukishova, Prof. Boyd, and myself. Within this project, I have

performed a comparative study of laser performances of cholesteric liquid crys-

tals (CLCs) doped with two different laser dyes. One of the dyes is a well-known

DCM, broadly used in CLC structures. The other dye, oligofluorene (OF), is

a new laser dye with a high order parameter, synthesized in Prof. Shaw Horng

Chen’s group. I have experimentally shown that the new OF dye has a bet-

ter laser performance in both transverse single-mode and multi-mode regimes,

demonstrating higher total lasing output energy, better stability, and, under


certain conditions, higher slope efficiency [157]. More research in this area was

done in Prof. Chen’s group [156].

This work is described in detail in Chapter 4.

4. “Optical activity in diffraction from artificial planar chiral structures”

This project is a close collaboration between Dr. Sergei Volkov, Prof. Boyd’s

group, and the groups of Profs. Martti Kauranen, Jari Turunen, and Yuri Svirko

from Finland. There have been many controversies in the literature regarding

the optical activity in artificial chiral structures. Our study was aimed at re-

solving the contradictions.

My role was to perform an experimental part of the work, namely, the mea-

surement of the polarization azimuth rotation and ellipticity of light, diffracted

from planar structures with pure structural chirality. The measurements were

also performed for the patterns with molecular chirality. The observed effects

were similar for both types of patterns. The fact that the molecular and pure

structural chirality manifest themselves similarly in diffraction invites the con-

clusion that it does not matter what is the origin of chirality in the sample.

What matters is that both molecularly- and structurally-chiral patterns make

the diffraction experimental setup chiral, which results in similar optical ac-

tivity effects from both types of patterns. The statement was supported by a

simple wire model that gave not only qualitative, but also a good quantitative

description to my experimental data.


This work is described in Chapter 5. It gave a material for a publication [169].

I believe that the work that I have performed during my Ph. D. program will

contribute to the development of new materials with improved optical properties,

and to the conceptual understanding of some aspects of linear and nonlinear optics.

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Appendix A

Mesoscopic Field in an Inclusion of

a Maxwell Garnett Composite

Material

In this Appendix we derive the expression for the mesoscopic field in an inclusion of

a Maxwell Garnett composite material. As a starting point, we use the result for the

mesoscopic field e(r) at any point r of a Maxwell Garnett composite, obtained in [25]

(Eq. (3.8)):

e(r) = E(r) +

∫

↔

T(r − r′)p′(r′) dr′ +4π

3ǫh

[P′(r) − p′(r)] . (A.1)

Here↔

T(r) designates a static dipole-dipole coupling tensor for a host medium with

dielectric constant ǫh. The mesoscopic polarization p′(r) is a linear part of the source

178

APPENDIX A. MAXWELL GARNETT MESOSCOPIC ELECTRIC FIELD 179

polarization defined in [25] (we will refer to it as the linear source polarization). It is

defined as

p′(r) =

(χ(1)i − χ

(1)h )e(r), if r ∈ inclusion,

0, if r ∈ host.

(A.2)

Here χ(1)i and χ

(1)h are the susceptibilities of the inclusion and host media, respectively.

In this study we are not concerned with the nonlinear interactions, so we do not

consider the nonlinear part of the source polarization. The macroscopic linear source

polarization P′(r) is obtained by averaging p′(r),

P′(r) =

∫

∆(r − r′)p′(r′) dr′, (A.3)

where ∆(r) is a smoothly varying weighting function which has a range R much

smaller than the wavelength of light, but much larger than a typical separation dis-

tance between the inclusions. The weighting function is normalized to unity:

∫

∆(r − r′) dr′ = 1. (A.4)

We are considering the mesoscopic field in an inclusion of the Maxwell Garnett

composite material. In the case of an isotropic and uniform inclusion material one

can assume that the polarization p′(r) and the electric field e(r) are mesoscopically

uniform over an inclusion. Based on the above assumption and on the mathematical

arguments given in Ref. [25], we can set the term involving the dipole-dipole coupling

APPENDIX A. MAXWELL GARNETT MESOSCOPIC ELECTRIC FIELD 180

tensor in Eq. (A.1) equal to zero. It brings us to the expression

ei(r) = E(r) +4π

3ǫh

[P′(r) − p′(r)] (A.5)

for the mesoscopic field in an inclusion. We can find the macroscopic average po-

larization P′(r) from Eq. (A.3), using the assumption that p′(r) is mesoscopically

uniform over an inclusion:

P′(r) = fip′(r). (A.6)

Here fi is the volume fraction of the inclusions in the composite material. Using

Eq. (A.2), we can express the mesoscopic polarization pi(r) in an inclusion as

pi(r) = χ(1)h ei(r) + p′(r). (A.7)

Substituting Eqs. (A.6) and (A.7) into Eq. (A.5) and making use of the relation

fh +fi = 1 for the volume fractions of the host and inclusion materials, we obtain the

expression

ei(r) =3ǫh


[

E(r) − 4π

3ǫh

fh pi(r)

]

(A.8)

for the mesoscopic field in an inclusion.

Appendix B

Multiple Solutions for the

Population Inversion

Let us consider in which part of parameter space Eq. (3.1a) has more than one physical

solution. It is convenient to rewrite the equation in the form

w[1 + (δ + δLw)2 + x] = −[1 + (δ + δLw)2], (B.1)

where δ = ∆T2 is the detuning parameter, δL = ∆LT2 is the Lorentz red-shift pa-

rameter, and x = |E|2/|E0s |2 is the electric field parameter. We rewrite Eq. (B.1)

as [170]

w3 + a2w2 + a1w + a0 = 0, (B.2)

181

APPENDIX B. OPTICAL BISTABILITY 182

where

a0 =1 + δ2

δ2L

, (B.3a)

a1 =1 + δ2 + x + 2δδL

δ2L

, (B.3b)

and

a2 =2δ + δL

δL

. (B.3c)

Then, constructing

q =a1

3− a2

2

9(B.4a)

and

r =a1a2 − 3a0

6− a3

2

27, (B.4b)

we look at the sign of the parameter

D = q3 + r2. (B.5)

If D > 0, there is one real root and a pair of complex conjugate roots; if D = 0 all

roots are real and at least two are equal; if D < 0 all three roots are real (irreducible

case) [170]. In order to achieve multiple physical solutions, we need the values of w

to be real (and in the range −1 to +1). Certainly a necessary condition for this is

D ≤ 0. Introducing

s1 =[

r + (q3 + r2)1

2

]1

3

(B.6a)


and

s2 =[

r − (q3 + r2)1

2

]1

3

, (B.6b)

we write the solutions to Eq. (B.2) in the form [170]

w1 = (s1 + s2) −a2

3, (B.7a)

w2 = −1

2(s1 + s2) −

a2

3+

i√

3

2(s1 − s2), (B.7b)

and

w3 = −1

2(s1 + s2) −

a2

3− i

√3

2(s1 − s2). (B.7c)

We now consider certain fixed values of the red shift parameter δL, and investigate

the ranges of δ and x for which multiple physical solutions exist. Such ranges are

marked with contours on the graphs in Fig. B.1. For the values of parameters δ and

x lying inside the contours there are three physical solutions to Eq. (B.2) with the

corresponding values of w being within the range [−1 : 1]. The area outside the

contours corresponds to a single physical solution for w, with the other two solutions

being complex and, therefore, non-physical. According to our numerical analysis,

summarized in Fig. B.1, multiple physical solutions only arise for |δL| ≥ 4.16. In

sodium vapor that we consider as an example for our analysis in Section 3.3, such

large values of δL are not achievable; raising the density to increase ∆L also decreases

T2 due to homogeneous broadening, and δL can never get this large.


-8

-7

-6

-5

-4

-3

-2

-1

0

1

0.1 1 10 100

δ

log x

δL = −10

δL = −9

δL = −8δL = −7

δL = −6

δL = −5

δL = −4.16

Figure B.1: The ranges of values of the detuning parameter δ and the electric

field parameter x for which mirrorless optical bistability is achievable (marked with

contours). Different contours correspond to different values of the Lorentz red-

shift parameter δL. The point corresponds to the limiting value δL = −4.16: for

|δL| < 4.16 there are no ranges of δ and x at which optical bistability is achievable.

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